%%Creator: %!PS-Adobe-2.0 dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Math-255.dvi %%Pages: 97 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips Math-255 %DVIPSParameters: dpi=300 %DVIPSSource: TeX output 1999.12.14:1213 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 40258431 52099146 1200 300 300 (Math-255.dvi) @start
%DVIPSBitmapFont: Fa eufm10 6 1 /Fa 1 113 df<4300E7806BC070C060C060C060C060C060C060C0FD807F006400600060 00600060000A117F8B0C>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmbx5 6 2 /Fb 2 91 df<01FF0807FFF81F00F83C0038780018780018F00000F00000F00000F00000 F000007800187800183C00301F00F007FFC001FF0015117E901B>67 D<FFFF80FFFF80F01F00E01E00C03C0000780000F80001F00003E00007C0000F80000F01 801E01803C03807C0F00FFFF00FFFF0011117E9017>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmmi5 6 31 /Fc 31 123 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmr5 6 16 /Fd 16 116 df<020408103020604040C0C0C0C0C0C0C0C0404060203010080402071A7D 920D>40 D<8040201018080C0404060606060606060604040C081810204080071A7D920D >I<0F80306060304010C018C018C018C018C018C018C018C0186030603030600F800D10 7E8F11>48 D<0C003C00CC000C000C000C000C000C000C000C000C000C000C000C000C00 FFC00A107D8F11>I<3FC06060F030F0386018003800380030006000C003000600080830 187FF0FFF00D107E8F11>I<FFFFF0FFFFF0000000000000000000000000FFFFF0FFFFF0
14087D891A>61 D<FFFC001C07001C01801C01C01C01C01C01C01C01801C07001FFC001C 00001C00001C00001C00001C00001C00001C0000FF800012117F9016>80 D<FCFCC0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0FCFC06197D920A>91 D<FCFC0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0CFCFC06197F920A>93 D<0FC030E060E0C000C000C000C000C000602030400F800B0B7E8A0F>99 D<0078001800180018001800180F98307860184018C018C018C0184018603830780F9E0F 117E9013>I<0F0030C06060C060FFE0C000C000C000602030400F800B0B7E8A0F>I<F000 30003000300030003000307C30303040318033003F80318030C030603070FC7C0E117E90 12>107 D<F0303030303030303030303030303030FC06117E900A>I<0F8030606030C018 C018C018C018C018603030600F800D0B7E8A11>111 D<3E804180808080807C001F8000 C080408040E0809F000A0B7E8A0E>115 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe eufm10 8.4 3 /Fe 3 113 df<07E01E001FF07F0030F8C780607D8380603D0380E03E03E0F01E01C0F0 1E0080781E01007C0E03003C0E07001C0E0700180E0700100E0700600E0700000E070000 0E0700000C0700000C07000018078000180780062007900FC003E00F8003C0000001001C 197F971D>78 D<187078007CFCFE00FFFF7E001E3C1E001C1C0E001C1C0E001C1C0E001C 1C0E001C1C0E001C1C0E001C1C0E001C1C0E001C1C0F201E1E1FC01E1F0F800C1C060008 0800001B11808F1B>109 D<10002000604070F079FC7BFC3C3C381C381C381C381C381C 381C381C381C38187F38BFF03BE038803800380038003800380030000E1A7F9211>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff eufm10 12 5 /Ff 5 114 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmbx7 8.4 10 /Fg 10 114 df<01C007C0FFC0FBC003C003C003C003C003C003C003C003C003C003C003 C003C003C003C003C003C07FFF7FFF10167D9517>49 D<0FF0003FFE00783F00FC1F80FC 0FC0FC0FC07807C00007C0000FC0000F80001F00003E0000780000F00001C0000380C006 00C01C01C03FFF807FFF80FFFF80FFFF8012167E9517>I<0007000000070000000F8000 000F8000001FC000001FC0000033E0000033E0000073F0000061F00000E1F80000C0F800 01C0FC0001807C0001807C0003FFFE0003FFFE0006001F0006001F000C000F800C000F80 180007C0FF003FF8FF003FF81D187E9722>65 D<FFF807F8FFF807F80F8001800F800700 0F800E000F8018000F8030000F80E0000F81C0000F8300000F8F80000F9FC0000FB3E000 0FE1F0000FC0F8000F80FC000F807C000F803E000F801F000F800F800F8007C00F8003E0 FFF81FFCFFF81FFC1E187E9723>75 D<FFF800FFF8000F80000F80000F80000F80000F80 000F80000F80000F80000F80000F80000F80000F80000F80000F800C0F800C0F801C0F80 1C0F80380F80380F80F8FFFFF8FFFFF816187E971B>I<FFFFF000FFFFFC000F803F000F 801F800F800F800F800FC00F800FC00F800FC00F800FC00F800F800F801F800F803F000F FFFC000FFFF0000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F 800000FFF80000FFF800001A187E971F>80 D<003FE00001FFFC0007E03F000F800F801F 0007C03E0003E07E0003F07E0003F07C0001F0FC0001F8FC0001F8FC0001F8FC0001F8FC
0001F8FC0001F8FC0001F87E0003F07E0003F03E0F03E01F1FC7C00FB0EF8007F07F0001 FFFC00003FF81800003C1800003FF800001FF800001FF000000FF0000007E0000003801D 1F7E9722>I<7FFFFE7FFFFE7E007C7800F87001F07003F06007E0600FC0000F80001F00 003E00007E0000FC0000F80001F00603E00607E0060FC00E1F800E1F001C3E003C7C00FC FFFFFCFFFFFC17187E971C>90 D<FC7F00FFFFC03F03E03C01F03C01F03C00F83C00F83C 00F83C00F83C00F83C01F03E01F03F07E03DFFC03C7E003C00003C00003C00003C00003C 0000FF0000FF000015167E8E19>112 D<03F0601FFCE03F06E07C03E07C01E0F801E0F8 01E0F801E0F801E0F801E07C01E07C03E03E0FE01FFDE007F1E00001E00001E00001E000 01E00001E00007F80007F815167F8E18>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmex10 12 47 /Fh 47 126 df<000800100020004000C0018003000300060006000C000C001C00180038 003800300070007000700070006000E000E000E000E000E000E000E000E000E000E000E0 00E000E000E000E0006000700070007000700030003800380018001C000C000C00060006 0003000300018000C000400020001000080D3B798117>0 D<800040002000100018000C 0006000600030003000180018001C000C000E000E0006000700070007000700030003800 380038003800380038003800380038003800380038003800380038003000700070007000 70006000E000E000C001C00180018003000300060006000C00180010002000400080000D 3B7E8117>I<0000C0000700001C0000380000700000600000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00001C00001C0000380000F0000380000C000003800000F000003800001C000 01C00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E000006000007000003800001C00000700 0000C0123B7B811D>8 D<C000003800000F000003800001C00001C00000E00000E00000 E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000 E00000E00000E00000E000006000007000003800001C000007000000C0000700001C0000 380000700000600000E00000E00000E00000E00000E00000E00000E00000E00000E00000 E00000E00000E00000E00000E00000E00000E00000E00000E00001C00001C0000380000F 0000380000C00000123B7B811D>I<C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0 C0C0C0C0C0C0C0C0C0C0C00220798011>12 D<0000600000C0000180000300000600000E 00001C0000180000380000700000600000E00001C00001C0000380000380000780000700 000700000F00000E00001E00001E00001C00001C00003C00003C00003800003800007800 00780000780000780000780000700000F00000F00000F00000F00000F00000F00000F000 00F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F000 007000007800007800007800007800007800003800003800003C00003C00001C00001C00 001E00001E00000E00000F000007000007000007800003800003800001C00001C00000E0 00006000007000003800001800001C00000E000006000003000001800000C00000601359 77811E>16 D<C000006000003000001800000C00000E000007000003000003800001C000 00C00000E000007000007000003800003800003C00001C00001C00001E00000E00000F00 000F000007000007000007800007800003800003800003C00003C00003C00003C00003C0 0001C00001E00001E00001E00001E00001E00001E00001E00001E00001E00001E00001E0 0001E00001E00001E00001E00001E00001E00001E00001E00001C00003C00003C00003C0 0003C00003C0000380000380000780000780000700000700000F00000F00000E00001E00 001C00001C00003C0000380000380000700000700000E00000C00001C000038000030000 0700000E00000C0000180000300000600000C0000013597F811E>I<0000018000000300 0000060000000C000000180000003000000060000000E0000001C0000001800000038000 00070000000E0000000E0000001C0000001C000000380000007800000070000000F00000 00E0000001E0000001C0000001C0000003C0000003800000078000000780000007000000 0F0000000F0000000E0000001E0000001E0000001E0000001C0000003C0000003C000000 3C0000003C00000038000000780000007800000078000000780000007800000078000000 7800000070000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000 F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000 F0000000F0000000F0000000F0000000F000000070000000780000007800000078000000 78000000780000007800000078000000380000003C0000003C0000003C0000003C000000 1C0000001E0000001E0000001E0000000E0000000F0000000F0000000700000007800000 078000000380000003C0000001C0000001C0000001E0000000E0000000F0000000700000 0078000000380000001C0000001C0000000E0000000E0000000700000003800000018000 0001C0000000E00000006000000030000000180000000C00000006000000030000000180 1977768125>I<C00000006000000030000000180000000C000000060000000300000003
80000001C0000000C0000000E00000007000000038000000380000001C0000001C000000 0E0000000F00000007000000078000000380000003C0000001C0000001C0000001E00000 00E0000000F0000000F0000000700000007800000078000000380000003C0000003C0000 003C0000001C0000001E0000001E0000001E0000001E0000000E0000000F0000000F0000 000F0000000F0000000F0000000F0000000F000000070000000780000007800000078000 000780000007800000078000000780000007800000078000000780000007800000078000 000780000007800000078000000780000007800000078000000780000007800000078000 00070000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000E0000 001E0000001E0000001E0000001E0000001C0000003C0000003C0000003C000000380000 00780000007800000070000000F0000000F0000000E0000001E0000001C0000001C00000 03C000000380000007800000070000000F0000000E0000001C0000001C00000038000000 3800000070000000E0000000C0000001C000000380000003000000060000000C00000018 0000003000000060000000C000000019777F8125>I<00000700001F00003C0000F80001 E00003C0000780000780000F00001E00001E00001E00003C00003C00003C00003C00003C 00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C 00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C 00003C00003C00003C00003C00003C00003C00003C0000780000780000700000F00000E0 0001C00003C0000780000E00003C0000780000E000007800003C00000E000007800003C0 0001C00000E00000F000007000007800007800003C00003C00003C00003C00003C00003C 00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C 00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C 00003C00003C00003C00003C00003C00003C00001E00001E00001E00000F000007800007 800003C00001E00000F800003C00001F00000718777A8125>26 D<000000000010000000 0000380000000000380000000000700000000000700000000000700000000000E0000000 0000E00000000001C00000000001C00000000001C0000000000380000000000380000000 000380000000000700000000000700000000000700000000000E00000000000E00000000 001C00000000001C00000000001C00000000003800000000003800000000003800000000 00700000000000700000000000700000000000E00000000000E00000000001C000000000 01C00000000001C000000000038000000000038000000000038000000000070000000000 0700000000000700000000000E00000000000E00000000000E00000000001C0000000000 1C0000000000380000000000380000000000380000000000700000000000700000000000 700000000000E00000000000E00000000000E00000000001C00000000001C00000000003 80000000000380000000000380000000000700000000000700000000000700000000000E 00000000000E00000000000E00000000001C00000000001C000000000038000000000038 0000000000380000000000700000000000700000000000700000000000E00000000000E0 0000000000E00000000001C00000000001C0000000000380000000000380000000000380 000000000700000000000700000000000700000000000E00000000000E00000000000E00 000000001C00000000001C00000000001C00000000003800000000003800000000007000 00000000700000000000700000000000E00000000000E00000000000E00000000001C000 00000001C00000000001C000000000038000000000038000000000070000000000070000 0000000700000000000E00000000000E00000000000E00000000001C00000000001C0000 0000001C0000000000380000000000380000000000700000000000700000000000700000 000000E00000000000E000000000004000000000002D777D8134>30 D<000000C000000180000003000000070000000E0000000C000000180000003800000070 00000060000000E0000001C0000001C000000380000007800000070000000F0000000E00 00001E0000001C0000003C000000380000007800000078000000F0000000F0000000F000 0001E0000001E0000001E0000003C0000003C0000003C000000780000007800000078000 000F8000000F0000000F0000000F0000001F0000001F0000001E0000001E0000001E0000 003E0000003E0000003E0000003E0000003C0000003C0000007C0000007C0000007C0000 007C0000007C0000007C0000007C0000007C00000078000000F8000000F8000000F80000 00F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F80000 00F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F80000 00F8000000F8000000F8000000F8000000F8000000F8000000F8000000780000007C0000 007C0000007C0000007C0000007C0000007C0000007C0000007C0000003C0000003C0000 003E0000003E0000003E0000003E0000001E0000001E0000001E0000001F0000001F0000 000F0000000F0000000F0000000F80000007800000078000000780000003C0000003C000 0003C0000001E0000001E0000001E0000000F0000000F0000000F0000000780000007800 0000380000003C0000001C0000001E0000000E0000000F00000007000000078000000380 000001C0000001C0000000E0000000600000007000000038000000180000000C0000000E
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
003F003F003F003F003F003F003F003F003F003F003F003F102D777E2C>I<FC00FC00FC 00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC 00FC00FC00FC00FC00FC00FC00FC007C007E007E003E003E001F001F000F800F8007C003 C001E000F00078003C001E00060F2D6D802C>I<003F003F003F003F003F003F003F003F 003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F 003F003E007E007E007E007C00FC00F801F801F003E003E007C00F801F003E007800F000 C000102D77802C>I<003F003F003F003F003F003F003F003F003F003F003F003F003F00 3F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003E007E007E00 7E007C00FC00F800F801F001E003E003C007800F001E003C007800E000E00078003C001E 000F00078003C003E001E001F000F800F800FC007C007E007E007E003E003F003F003F00 3F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F00 3F003F003F003F003F003F003F105C77802C>I<FC00FC00FC00FC00FC00FC00FC00FC00 FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00 FC00FC007C007E007E007E003E003E001F001F000F000F80078003C003E001E000F00078 001C000E000E001C007800F001E003E003C007800F800F001F001F003E003E007E007E00 7E007C00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00 FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC000F5C6D802C>I<FCFCFCFCFC FCFCFCFCFCFCFCFCFCFCFCFC06116D802C>I<C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0 C0C0C0C0C0C0C0C0C0C0C0C0C0C0C00220718021>I<FC000000FC000000FC000000FC00 0000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC00 0000FC000000FC000000FC000000FC0000007C0000007C0000007C0000007E0000007E00 00007E0000007E0000007E0000007E0000007E0000007E0000003E0000003E0000003E00 00003E0000003F0000003F0000003F0000003F0000001F0000001F0000001F0000001F80 00001F8000000F8000000F8000000F8000000F8000000FC0000007C0000007C0000007C0 000007C0000003E0000003E0000003E0000001E0000001F0000001F0000000F0000000F0 000000F800000078000000780000007C0000003C0000003C0000001E0000001E0000001E 0000000F0000000F000000078000000780000003C0000003C0000001C0000001E0000000 E00000007000000070000000380000003C0000001C0000000E0000000E00000007000000 038000000180000001C0000000E000000060000000301C5A72822C>I<000003F0000003 F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003 F0000003F0000003F0000003F0000003F0000003F0000003F0000003E0000003E0000003 E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007 C0000007C0000007C0000007C000000FC000000FC000000FC000000FC000000F8000000F 8000000F8000001F8000001F8000001F0000001F0000001F0000001F0000003F0000003E 0000003E0000003E0000003E0000007C0000007C0000007C00000078000000F8000000F8 000000F0000000F0000001F0000001E0000001E0000003E0000003C0000003C000000780 000007800000078000000F0000000F0000001E0000001E0000003C0000003C0000003800 00007800000070000000E0000000E0000001C0000003C000000380000007000000070000 000E0000001C00000018000000380000007000000060000000C00000001C5A7F822C>I< FCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFC06207280 2C>I<FCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFCFC06 2069802C>I<00001FF800000000FFFF00000003FFFFC000000FE1C7F000003F01C0FC00 007C01C03E0000F001C00F0001E001C0078003C001C003C0078001C001E0070001C000E0 0E0001C000701E0001C000781C0001C00038380001C0001C380001C0001C380001C0001C 700001C0000E700001C0000E700001C0000EE00001C00007E00001C00007E00001C00007 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE00001C00007E00001C00007E00001C00007 E00001C00007700001C0000E700001C0000E700001C0000E380001C0001C380001C0001C 380001C0001C1C0001C000381E0001C000780E0001C00070070001C000E0078001C001E0 03C001C003C001E001C0078000F001C00F00007C01C03E00003F01C0FC00000FE1C7F000 0003FFFFC0000000FFFF000000001FF8000030327D7F37>76 D<0000000FFF0000000000 0000FFFFF0000000000007FFFFFE00000000001FFFFFFF80000000007FF8F1FFE0000000 00FF80F01FF000000003FC00F003FC00000007F000F000FE0000000FC000F0003F000000 1F8000F0001F8000003E0000F00007C000007C0000F00003E00000F80000F00001F00001 F00000F00000F80001E00000F00000780003E00000F000007C0007C00000F000003E0007 800000F000001E000F800000F000001F001F000000F000000F801E000000F0000007801E 000000F0000007803E000000F0000007C03C000000F0000003C03C000000F0000003C07C 000000F0000003E078000000F0000001E078000000F0000001E078000000F0000001E0F8 000000F0000001F0F0000000F0000000F0F0000000F0000000F0F0000000F0000000F0FF FFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFF0F0
000000F0000000F0F0000000F0000000F0F0000000F0000000F0F8000000F0000001F078 000000F0000001E078000000F0000001E078000000F0000001E07C000000F0000003E03C 000000F0000003C03C000000F0000003C03E000000F0000007C01E000000F0000007801E 000000F0000007801F000000F000000F800F800000F000001F0007800000F000001E0007 C00000F000003E0003E00000F000007C0001E00000F00000780001F00000F00000F80000 F80000F00001F000007C0000F00003E000003E0000F00007C000001F8000F0001F800000 0FC000F0003F00000007F000F000FE00000003FC00F003FC00000000FF80F01FF0000000 007FF8F1FFE0000000001FFFFFFF800000000007FFFFFE000000000000FFFFF000000000 00000FFF0000000044467D7F4B>I<00001FF800000000FFFF00000003FFFFC000000FE0 07F000003F0000FC00007C00003E0000F000000F0001E00000078003C0000003C007E000 0007E007F000000FE00E7800001E701E3C00003C781C1E00007838380F0000F01C380780 01E01C3803C003C01C7001E007800E7000F00F000E7000781E000EE0003C3C0007E0001E 780007E0000FF00007E00007E00007E00003C00007E00003C00007E00007E00007E0000F F00007E0001E780007E0003C3C00077000781E000E7000F00F000E7001E007800E3803C0 03C01C38078001E01C380F0000F01C1C1E000078381E3C00003C780E7800001E7007F000 000FE007E0000007E003C0000003C001E00000078000F000000F00007C00003E00003F00 00FC00000FE007F0000003FFFFC0000000FFFF000000001FF8000030327D7F37>I<FFFF FFFFFFC0FFFFFFFFFFC078000000FFE07C0000000FE03E00000003E01F00000000F00F00 000000700F800000003007C00000001803C00000000803E00000000801F00000000400F8 00000000007800000000007C00000000003E00000000001E00000000000F00000000000F 800000000007C00000000003C00000000003E00000000001F00000000000F80000000000 780000000000780000000000300000000000700000000000600000000000C00000000001 80000000000300000000000700000000000600000000000C000000000018000000000038 0000000000700000000400600000000C00C0000000080180000000180380000000380300 000000700600000000F00C00000001F01800000007E038000000FFE03FFFFFFFFFE07FFF FFFFFFC0FFFFFFFFFFC02E327D7F35>80 D<FFFFFFFFFFFFFFFFFFFF1FC00003F80FC000 03F007C00003E007C00003E007C00003E007C00003E007C00003E007C00003E007C00003 E007C00003E007C00003E007C00003E007C00003E007C00003E007C00003E007C00003E0 07C00003E007C00003E007C00003E007C00003E007C00003E007C00003E007C00003E007 C00003E007C00003E007C00003E007C00003E007C00003E007C00003E007C00003E007C0 0003E007C00003E007C00003E007C00003E007C00003E007C00003E007C00003E007C000 03E007C00003E007C00003E007C00003E007C00003E007C00003E007C00003E00FE00007 F01FF0000FF8FFFE007FFFFFFE007FFF28327D7F2F>I<E0000000E0E0000000E0E00000 00E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000 E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0 E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0 000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E000 0000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E070000001C0700000 01C0380000038038000003801C000007001E00000F000F00001E000780003C0003E000F8 0001F803F000007FFFC000001FFF00000007FC000023327D7F2A>83 D<0007FC0000001FFF0000007FFFC00001F803F00003E000F8000780003C000F00001E00 1E00000F001C000007003800000380380000038070000001C070000001C0E0000000E0E0 000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E000 0000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E00000 00E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000 E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0 E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E0E0000000E023 327D7F2A>I<FFFFFFFFFFFFFFE000FFFFFFFFFFFFFFF0007F8000000007FFF0007FC000 0000003FF0003FC00000000007F8001FE00000000000F8000FF000000000007C000FF800 000000001C0007F800000000000E0003FC0000000000060003FE0000000000020001FE00 00000000010000FF00000000000100007F80000000000080007FC0000000000000003FC0 000000000000001FE0000000000000001FF0000000000000000FF00000000000000007F8 0000000000000003FC0000000000000003FC0000000000000001FE0000000000000000FF 0000000000000000FF80000000000000007F80000000000000003FC0000000000000003F E0000000000000001FE0000000000000000FF00000000000000007F80000000000000007 FC0000000000000003FC0000000000000001FE0000000000000001FF0000000000000000 FF00000000000000007E00000000000000003C0000000000000000380000000000000000 700000000000000000E00000000000000001E00000000000000001C00000000000000003 80000000000000000700000000000000000E00000000000000001C00000000000000003C
0000000000000000380000000000000000700000000000000000E00000000000000001C0 000000000000000380000000000000000780000000000000000700000000000080000E00 000000000100001C0000000000010000380000000000030000700000000000060000F000 000000000E0000E000000000001E0001C000000000003C0003800000000000FC00070000 00000003F8000E00000000001FF8001E0000000007FFF8001FFFFFFFFFFFFFF0003FFFFF FFFFFFFFF0007FFFFFFFFFFFFFF000FFFFFFFFFFFFFFE00041467D7F48>88 D<FFFFFFFFFFFFFF80FFFFFFFFFFFFFF80FFFFFFFFFFFFFF800FFF0000007FF80003FF00 00007FE00001FF0000007FC00000FF0000007F800000FF0000007F800000FF0000007F80 0000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF00 00007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F80 0000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF00 00007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F80 0000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF00 00007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F80 0000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF00 00007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F80 0000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF00 00007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F80 0000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF00 00007F800000FF0000007F800000FF0000007F800000FF0000007F800000FF0000007F80 0000FF0000007F800001FF800000FFC00003FFC00001FFE0000FFFF00007FFF800FFFFFF 007FFFFF80FFFFFF007FFFFF80FFFFFF007FFFFF8039467D7F40>I<600000000006F000 0000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000 0000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000 0000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000 0000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000 0000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000 0000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000 0000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000 0000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF000 0000000FF8000000001F78000000001E78000000001E78000000001E7C000000003E3C00 0000003C3E000000007C1F00000000F80F00000000F00F80000001F007C0000003E003E0 000007C001F800001F8000FC00003F00007F0000FE00003FF00FFC00000FFFFFF0000007 FFFFE0000001FFFF800000001FF8000030467D7F37>91 D<00001FF800000001FFFF8000 0007FFFFE000000FFFFFF000003FF00FFC00007F0000FE0000FC00003F0001F800001F80 03E0000007C007C0000003E00F80000001F00F00000000F01F00000000F83E000000007C 3C000000003C7C000000003E78000000001E78000000001E78000000001EF8000000001F F0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000F F0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000F F0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000F F0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000F F0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000F F0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000F F0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000F F0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000F F0000000000F60000000000630467D7F37>I<000001C000000000000FF800000000007F FF0000000001FC1FC00000000FC001F80000007E00003F000003E0000003E0001F000000 007C00F000000000078080000000000080310A80A532>99 D<0000000000040000000000 0C0000000000180000000000180000000000300000000000300000000000600000000000 600000000000C00000000000C00000000001800000000001800000000003000000000003 00000000000600000000000600000000000600000000000C00000000000C000000000018 0000000000180000000000300000000000300000000000600000000000600000000000C0 0000000000C0000000000180000000000180000000000300000200000300000700000600 000F00000600003780000C00006780000C000087800018000003C00018000003C0003000 0001E00030000001E00060000000F00060000000F000C0000000F000C000000078018000 00007801800000003C01800000003C03000000001E03000000001E06000000001E060000 00000F0C000000000F0C0000000007980000000007980000000003F00000000003F00000 000003E00000000001E00000000001C00000000000C00000002E3C7B8132>112 D<00300000780000780000FC0001B6000733800E31C03C30F0F0303CC0300C0030000030
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ndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmsy5 6 12 /Fi 12 106 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsy7 8.4 35 /Fj 35 113 df<FFFFFEFFFFFE17027C891F>0 D<60F0F06004047C8A0C>I<8000C0C001 C0600300300600180C000C180006300003600001C00001C0000360000630000C1800180C 00300600600300C001C08000C0121279911F>I<0300030003004308F33C3B700FC00300 0FC03B70F33C43080300030003000E0F7D8F14>I<007F00000188C00006083000080808 001008040020080200200802004008010040080100800800808008008080080080FFFFFF 808008008080080080800800808008008040080100400801002008020020080200100804 0008080800060830000188C000007F0000191A7D951F>8 D<007F00000180C000060030 000800080018000C00240012002200220041004100408081008041008080220080801400 808008008080080080801400808022008080410080408081004100410022002200240012 0018000C0008000800060030000180C000007F0000191A7D951F>10 D<FFFFFF807FFFFF80000000000000000000000000000000000000000000000000FFFFFF 80FFFFFF800000000000000000000000000000000000000000000000007FFFFF80FFFFFF 8019127D911F>17 D<007FFE03FFFE0780000C0000180000300000300000600000600000 C00000C00000C00000C00000C00000C000006000006000003000003000001800000C0000 07800003FFFE007FFE0000000000000000000000000000000000007FFFFE7FFFFE17207C 971F>I<00000600001E0000780001E0000780000E0000380000F00003C0000F00003C00 00F00000F000003C00000F000003C00000E000003800001E000007800001E00000780000 1E0000060000000000000000000000000000000000007FFFFCFFFFFE17207C971F>20 D<C00000F000003C00000F000003C00000E000003800001E000007800001E00000780000 1E00001E0000780001E0000780000E0000380000F00003C0000F00003C0000700000C000 000000000000000000000000000000000000007FFFFCFFFFFE17207C971F>I<0F800080 3FE000807FF0008070780180C01C0180C00F07008007FF008003FE008000F80019097D8C 1F>24 D<018000000001800000000300000000030000000006000000000E000000001C00 0000003800000000FFFFFFFF80FFFFFFFF8038000000001C000000000E00000000060000 0000030000000003000000000180000000018000000021127C9128>32 D<000000C000000000C00000000060000000006000000000300000000038000000001C00 0000000E00FFFFFFFFC0FFFFFFFFC00000000E000000001C000000003800000000300000
000060000000006000000000C000000000C00022127D9128>I<0F8000803FE000807FF0 0080F07C0180C01F0780800FFF008003FE008000F8000000000000000000000000000000 00000000000000000000000000007FFFFF80FFFFFF8019117D901F>39 D<060F0F0F1E1E1C3C38383870706060E0C04008127E920B>48 D<07E000F8001FF8038E 00303C0E0100601E18008040073000808003E000408003C000408001E000408000F00040 8001F00040400338008040061E0180201C0F03001C7007FE0007C001F800220F7D8E28> I<007FE003FFE00780000C0000180000300000300000600000600000C00000C00000FFFF E0FFFFE0C00000C000006000006000003000003000001800000C000007800003FFE0007F E013187C941B>I<0000600000600000C0000180000180000300000300000600000C0000 0C0000180000180000300000600000600000C00000C00001800003000003000006000006 00000C0000180000180000300000300000600000C00000400000131E7A9700>54 D<C0C0C0C0C0E0E0C0C0C0C0C0030C7D8E00>I<400004C0000C60001860001830003030 00303000301800601FFFE00FFFC00C00C006018006018003030003030001860001860001 860000CC0000CC0000780000780000300000300016187F9718>I<000800000018000000 180000001800000018000000180000001800000018000000180000001800000018000000 180000001800000018000000180000001800000018000000180000001800000018000000 18000000180000FFFFFF80FFFFFF8019187C9720>63 D<000FC0003FC000C3C00103C006 03800E03801C07001C0600380000380000780000700000700000F00000F00000F00000F0 0000F00000F80080F803007C06007F08003FF0000FC00012187E9716>67 D<000FFFF8007FFFF800C380700383800007078000060780000C07000000070000000F00 00000E0000000E0000001FFF00001FFE00001C0000003800000038000000700000007000 0000E0000000E0000001C0000079800000FF0000007C0000001D187E971D>70 D<000FE0007FF000C1F00300F00600E00C00C01C01803803003800007800007000007000 60F000E0F001C0F001C0F003C0F803C0F807807C0F807E1F003FE7001F8E00000E00001C 001010003FE000FF8000141B7D9718>I<00C003C003C01FC00FC0318003C0C00003C300 0003C40000038800000390000007B0000007F0000007700000077800000F7800000E3C00 000E3C00001E1C00001C1E00001C1F00003C0F0040380780C03807C1807003E1007001FE 00C000F8001A187E971E>75 D<00107E000061FF0001861F80030C0F80061C07C00C3807 C01C3003C0387003C038C003C0700003C0700003C0F0000380F0000380F0000780F00007 00F0000700F8000E00F8000C00780018007C0030007E0040003F8380001FFE000007F000 001A187D9720>79 D<00FFFC0003FFFF000C3C3F8018380FC0703807C0607803C0C07803 C0007803C000780380007003800070070000F0060000F00C0000E0100000E0600001EF80 0001DE000001C0000003C0000003800000038000000780000007000000060000000C0000 001A197E971C>I<0007F0001FFC0020FC00C03C01C03803C02003C00003E00003F00001 F80000FF00007F80000FE00007E00001F01801F03000F07000F0F000E0F800C0FC01807F 06003FFC000FE00016187E9718>83 D<01FC000FFF801E03C0300060600030C00018C000 18C00018C00018C00018C00018C00018C00018C00018C00018C00018C00018C00018C000 1840000815147D931B>92 D<007001C00380070007000700070007000700070007000700 0700070007000E001C00F0001C000E000700070007000700070007000700070007000700 07000700038001C000700C237C9914>102 D<F0001C000E000700070007000700070007 00070007000700070007000700038001C0007001C0038007000700070007000700070007 00070007000700070007000E001C00F0000C237C9914>I<008001800300030006000600 06000C000C001800180030003000300060006000C000C000600060003000300030001800 18000C000C00060006000600030003000180008009227C9910>I<C000C0006000600030 0030003000180018000C000C000600060006000300030001800180030003000600060006 000C000C001800180030003000300060006000C000C00009227D9910>I<C0C0C0C0C0C0 C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C002237B990C>I< 000000010000000300000006000000060000000C00000018000000180000003000000030 0000006000000060000000C0000001800000018000000300000003000000060008000C00 3C000C00DC0018000E0018000E00300007003000070060000380C0000380C00001C18000 01C1800000E3000000E3000000760000007C0000003C0000003800000018000020237D81 21>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmmi7 8.4 66 /Fk 66 123 df<03F0000E1810180C10380E10700720600720E00740E00740E00780C007 00C00700E00700601F003063201F81C0140F7D8E1A>11 D<00E001FC020E020002000300 0300038001C001E001F006F01C78387870387038E038E038E038E030E070E06060E030C0 1F000F197E9812>14 D<004000400040007C00C2013C0200040008001800300020006000 6000C000C000C000C000C000C000E000F0007E003FC01FE003F0007000300030046003C0
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
7D8E12>I<03E00C100838183818301F001FE00FF003F00030E030E030C06040C03F000D 0F7D8E13>I<018003800380070007000700FFE00E000E000E000E001C001C001C001C00 380038403840388039000E000B157E940F>I<1C0600660E00460E008E0E008E0E000E1C 001C1C001C1C001C1C0038380038388038388018390018F9000F0E00110F7D8E18>I<1C 0C661E460E8E068E060E041C041C041C0838083808381018201C4007800F0F7D8E14>I< 078F8008D0C010E1C020E18020E00001C00001C00001C00001C000038000038100C38100 E78200C5840078F800120F7E8E17>120 D<1C06660E460E8E0E8E0E0E1C1C1C1C1C1C1C 383838383838183818F00F700070006060E0E0C0C18043003C000F167D8E14>I<07020F 841FF81010002000400080010002000C00100820183FF047E081C00F0F7D8E13>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmr7 8.4 45 /Fl 45 119 df<01FE000703801C00E0380070780078700038F0003CF0003CF0003CF000 3CF0003C7800787800783800701C00E01C00E00C00C0060180860184820104C3030C7F03 F87F03F87F03F816187E971C>10 D<007E0003C1000703800E03800E01000E00000E0000 0E00000E0000FFFF800E03800E03800E03800E03800E03800E03800E03800E03800E0380 0E03800E03800E03800E03807F8FF01418809717>12 D<FFF00C017D9314>22 D<000C0600000C0600000C060000180C0000180C0000180C0000180C0000180C00003018 0000301800FFFFFFE0FFFFFFE000603000006030000060300000C0600000C0600000C060 00FFFFFFE0FFFFFFE00180C0000301800003018000030180000301800006030000060300 00060300000C0600000C0600000C0600001B1F7E9721>35 D<00800100020004000C0018 00180030003000600060006000E000E000E000E000E000E000E000E000E000E000600060 00600030003000180018000C00040002000100008009227D9910>40 D<800040002000100018000C000C00060006000300030003000380038003800380038003 800380038003800380030003000300060006000C000C001800100020004000800009227E 9910>I<000C0000000C0000000C0000000C0000000C0000000C0000000C0000000C0000 000C0000000C0000000C0000000C0000FFFFFF80FFFFFF80000C0000000C0000000C0000 000C0000000C0000000C0000000C0000000C0000000C0000000C0000000C0000000C0000 191A7E951F>43 D<07E01C38300C300C60066006E007E007E007E007E007E007E007E007 E007E0076006700E300C381C1C3807E010167F9514>48 D<03000700FF00070007000700 070007000700070007000700070007000700070007000700070007000700FFF00C167D95 14>I<0FC0307040384038C03CE01CE01C003C003800380070006000C001800300060004 040804100C3FF87FF8FFF80E167E9514>I<0F8030E06070707870382078007800700060 00C00F8000E000700038003C403CE03CE03C8038407030E00F800E167E9514>I<003000 30007000F000F0017002700670047008701070107020704070C070FFFF00700070007000 70007007FF10167F9514>I<03F00608180C181C301C70006000E000E7C0E860F030F018 E01CE01CE01CE01C601C601C30183030186007C00E167E9514>54 D<40007FFF7FFE7FFCC00480088010002000400040008001800180010003000300030007 000700070007000700020010177E9614>I<60F0F0600000000000000060F0F060040F7D 8E0B>58 D<FFFFFF80FFFFFF800000000000000000000000000000000000000000000000 00FFFFFF80FFFFFF80190A7E8D1F>61 D<FFFF001C03C01C00E01C00381C00381C001C1C 001E1C000E1C000E1C000F1C000F1C000F1C000F1C000F1C000F1C000F1C000E1C001E1C 001C1C003C1C00381C00F01C03C0FFFF0018187E971E>68 D<003F020001C0C60007003E 000E000E001C000E0038000600380002007800020070000200F0000000F0000000F00000 00F0000000F001FFC0F0000E0070000E0078000E0038000E0038000E001C000E000E000E 0007001E0001C02600003FC2001A187E971F>71 D<FF807F1C003C1C00301C00201C0040 1C01801C02001C04001C08001C10001C38001C5C001C9C001D0E001E07001C07001C0380 1C01C01C00E01C00E01C00701C00381C007CFF80FF18187E971E>75 D<FFFF001C03C01C00E01C00701C00781C00781C00781C00781C00781C00701C00E01C03 C01FFF001C00001C00001C00001C00001C00001C00001C00001C00001C00001C0000FF80 0015187E971B>80 D<7FFFFE701C0E401C02401C02C01C03801C01801C01801C01001C00 001C00001C00001C00001C00001C00001C00001C00001C00001C00001C00001C00001C00 001C00001C0003FFE018187F971C>84 D<FF80FE1C00381C00101C00101C00101C00101C 00101C00101C00101C00101C00101C00101C00101C00101C00101C00101C00101C00101C 00100E00200600600300C001C180007E0017187E971D>I<FF803F801C000E001C000400 0E0008000E0008000E0008000700100007001000038020000380200003C0600001C04000 01C0400000E0800000E0800000F180000071000000710000003A0000003A0000001C0000 001C0000001C00000008000019187F971D>I<FCFCC0C0C0C0C0C0C0C0C0C0C0C0C0C0C0 C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0FCFC06237D990B>91 D<FCFC0C0C0C0C0C0C0C0C
0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0CFCFC062380990B>93 D<1824428108047B9614>I<3F8070C070602070007000700FF038707070E070E071E071 E0F171FB1E3C100F7E8E14>97 D<07F81C1C301C70086000E000E000E000E000E0006000 700438081C1807E00E0F7F8E12>99 D<003E00000E00000E00000E00000E00000E00000E 00000E00000E0007CE000C3E00380E00700E00600E00E00E00E00E00E00E00E00E00E00E 00600E00700E00301E001C2E0007CF8011187F9716>I<07C0183030187018600CE00CFF FCE000E000E0006000700430081C1807E00E0F7F8E12>I<03E006700E701C201C001C00 1C001C001C00FF801C001C001C001C001C001C001C001C001C001C001C001C001C007F80 0C1880970C>I<0FCF001871803030007038007038007038007038003030001860002FC0 002000003000003FF0003FFC001FFE00600F00C00300C00300C00300C00300600600381C 0007E00011177F8E14>I<F8003800380038003800380038003800380039F03A183C1C38 1C381C381C381C381C381C381C381C381C381C381CFE7F10187E9716>I<307878300000 000000F838383838383838383838383838FE07187F970B>I<F800380038003800380038 00380038003800383E383C38303860388039803B803DC039E038F038703878383C383EFE 7F10187E9715>107 D<F838383838383838383838383838383838383838383838FE0718 7F970B>I<FC7C1F001D8E63801E0781C01E0781C01C0701C01C0701C01C0701C01C0701 C01C0701C01C0701C01C0701C01C0701C01C0701C01C0701C0FF9FE7F81D0F7F8E21>I< F9F03A183C1C381C381C381C381C381C381C381C381C381C381C381CFE7F100F7E8E16> I<07E01C38300C700E6006E007E007E007E007E0076006700E300C1C3807E0100F7F8E14 >I<F9F0003E1C00380E0038070038070038038038038038038038038038038038070038 0700380E003E180039F000380000380000380000380000380000380000FE000011167E8E 16>I<F9C03AE03CE03C403800380038003800380038003800380038003800FF000B0F7E 8E10>114 D<1F2060E0C020C020E020FC007F803FE007E080708030C030C020F0608F80 0C0F7F8E10>I<0400040004000C001C003C00FFC01C001C001C001C001C001C001C001C 001C201C201C201C200E4007800B157F9410>I<F87C381C381C381C381C381C381C381C 381C381C381C381C383C185C0F9F100F7E8E16>I<FE0F803807003C06001C04001C0400 0E08000E080007100007100003A00003A00003E00001C00001C000008000110F7F8E15> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmsy10 12 63 /Fm 63 121 df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
003C000000F0000003C000000F0000003C000000F0000003C000000F0000003C000000F0 000003C000000F0000003C000000F0000000F00000003C0000000F00000003C0000000F0 0000003C0000000F00000003C0000000F00000003C0000000F00000003C0000000F00000 003C0000000C000000000000000000000000000000000000000000000000000000000000 00007FFFFFF8FFFFFFFC1E287C9F27>I<C0000000F00000003C0000000F00000003C000 0000F00000003C0000000F00000003C0000000F00000003C0000000F00000003C0000000 F00000003C0000003C000000F0000003C000000F0000003C000000F0000003C000000F00 00003C000000F0000003C000000F0000003C00000070000000C000000000000000000000 000000000000000000000000000000000000000000000000007FFFFFF8FFFFFFFC1E287C 9F27>I<07E000010FF800013FFC0001381E0003700780036003C006C001E00EC000781C 80003FFC80001FF0800007E0200B7D9127>24 D<07E000011FF800013FFE0001781F0003 60078003C001E006C000F81E80007FFC80001FF8800007E0000000000000000007E00001 1FF800013FFE0001781F000360078003C001E006C000F81E80007FFC80001FF8800007E0 20167D9727>I<000FFFFC007FFFFC01F0000003800000060000000C0000001800000030 000000300000006000000060000000C0000000C0000000C0000000C0000000C0000000C0 000000C0000000C000000060000000600000003000000030000000180000000C00000006 0000000380000001E00000007FFFFC001FFFFC1E1E7C9A27>I<FFFFC000FFFFF0000000 3C0000000E000000030000000180000000C0000000600000006000000030000000300000 001800000018000000180000001800000018000000180000001800000018000000300000 00300000006000000060000000C0000001800000030000000E0000007C00FFFFF000FFFF 80001D1E7B9A27>I<00600000000000600000000000600000000000600000000000C000 00000000C0000000000180000000000300000000000300000000000E00000000001C0000 000000380000000000FFFFFFFFFFC0FFFFFFFFFFC03800000000001C00000000000E0000 00000003000000000003000000000001800000000000C00000000000C000000000006000 0000000060000000000060000000000060000000002A1A7C9832>32 D<00000001800000000001800000000001800000000001800000000000C00000000000C0 00000000006000000000003000000000003000000000001C00000000000E000000000007 00FFFFFFFFFFE0FFFFFFFFFFE0000000000700000000000E00000000001C000000000030 0000000000300000000000600000000000C00000000000C0000000000180000000000180 000000000180000000000180002B1A7D9832>I<00100000300000300000300000780000 780000FC0000FC0001B6000333000E31C03C30F0F0303CC0300C00300000300000300000 300000300000300000300000300000300000300000300000300000300000300000300000 300000300000300000300000300000300000300000300000300000300000300000300000 3000003000003000003000162D7FA219>I<006000030000006000030000006000030000 00600003000000C00001800000C00001800001800000C000030000006000030000006000 0E00000038001C0000001C00380000000E00FFFFFFFFFFC0FFFFFFFFFFC0380000000E00 1C0000001C000E000000380003000000600003000000600001800000C00000C000018000 00C0000180000060000300000060000300000060000300000060000300002A1A7C9832> 36 D<C00000000000E000000000007000000000003800000000001C00000000000E0000 00000007000000000003800000000001C00000000000E000000000007000000000003800 000000001C00000000000E000000000007000000000003800000000001C00000000000E0 00000000007000000000003800000000001C00000000000E000000000007000000000003 800000000001C00000000000E000100000007000300000003800300000001C0060000000 0E00600000000700C00000000380C000000001C0C000000000E0C00000000070C0000000 0038C0000000001CC0000000000EC00000000007C00000000003C000000003FFC0000000 0FFFE00000003C00600000007000002C2C7DA032>38 D<07E000010FF800013FFC000138 1E0003700780036003C006C001E00EC000781C80003FFC80001FF0800007E00000000000 000000000000000000000000000000000000000000000000000000000000007FFFFFFFFF FFFFFF20167D9627>I<0000C00000000001800000000001800000000003000000000003 00000000000600000000000C00000000001800000000003FFFFFFFE000FFFFFFFFE001C0 000000000700000000003E0000000000F800000000003C00000000000F00000000000380 0000000000C000000000007FFFFFFFE0003FFFFFFFE0001800000000000C000000000006 000000000003000000000003000000000001800000000001800000000000C00000002B1C 7D9932>I<00000060000000000030000000000030000000000018000000000018000000 00000C0000000000060000000000030000FFFFFFFF8000FFFFFFFFE00000000000700000 0000001C00000000000F800000000003E0000000000780000000001E0000000000380000 0000006000FFFFFFFFC000FFFFFFFF80000000000300000000000600000000000C000000 00001800000000001800000000003000000000003000000000006000002B1C7D9932>I< 000600030000000600030000000C00018000000C0001800000180000C00000380000E000
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ndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmmi10 12 79 /Fn 79 126 df<003F000000E180000380E020070060400E0070401C0030403C0038803C 003880780039007800390078003A00F0003A00F0003C00F0003800F00038007000380070 0078003000B800380318401C1C188007E007001B157E9420>11 D<00003E000000C18000 0300C0000400C0000800E0001000E0001000E0002000E0004000E0004001C0008001C000 80018000800380010007000101FE0001021C000103EE0002000700020007000200078002 0007800400078004000780040007800400078008000F0008000F0008000E0008001E0014 001C0014003800120070001300E0002081C000207E000020000000200000004000000040 0000004000000040000000800000008000000080000000800000001B2D7FA21C>I<000F 80003FF00040F000406000C00000C00000C00000C0000060000060000070000070000038 00003800001C00003E0000CE00038F000607000C07001C07803807803803807003807003 80700380E00300E00700E00700E00600600E00600C00300C003018001C300007C0001424 7EA316>14 D<000800001000001000001000000FF00008100037E0004000008000010000 0300000600000400000C0000180000180000300000300000600000600000600000C00000 C00000C00000C00000C00000C00000C00000C00000E00000E000007000007C00003F8000 1FE00007F80000FC00001E00000E0000060000060000060002040001080000F000142D7D A216>16 D<000F800018C000306000606000C07001C0700380700780700700700F00700E 00701E00701E00701C00F03C00F03C00F03FFFF07FFFE07FFFE07801E07801E07003C0F0 03C0F00380F00780F00700F00700F00E00700E00701C0070180030300030600018C0000F 000014237EA217>18 D<03800000C00000E000007000007000007800003800003800003C 00001C00001C00001E00000E00000E00000F000007000007000007800003800003800007 C00009C00011C00030E00060E000C0F00180700300700600780C00381C003838003C7000 1CE0001CC0000E17237DA21D>21 D<00C0030001C0070001C0070003800E0003800E0003 800E0003800E0007001C0007001C0007001C0007001C000E0038000E0038000E0038000E
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
0007800F8000078007C0000F0007C0000F0003C0000F0003E0000F0003E0001E0003C000 1E0007C0001E0007C0001E000F80003C000F00003C001E00003C007C00003C01F000007F FFE00000780078000078003C000078003E0000F0001E0000F0001E0000F0001F0000F000 1F0001E0001E0001E0003E0001E0003E0001E0007C0003C000780003C000F80003C003F0 0007C007C000FFFFFF8000FFFFFC000023227EA126>I<00007F00400003FFC080000FC0 6180001F001380007C000F8000F0000F0001E000070003C000070007C00007000F800006 000F000002001F000002003E000002003E000004007C000000007C000000007C00000000 F800000000F800000000F800000000F800000000F800000000F000000000F000001000F0 00001000F000002000F800002000F800004000780000C0007C000080003C000100001E00 0600000F000C000007C070000003FFE00000007F00000022247DA224>I<00FFFFF80000 FFFFFF000007801F8000078007C0000F0003E0000F0001E0000F0000F0000F0000F0001E 0000F0001E0000F8001E0000F8001E0000F8003C0000F8003C0000F8003C0000F8003C00 00F800780000F000780001F000780001F000780001F000F00003E000F00003E000F00003 C000F00007C001E000078001E0000F0001E0001F0001E0001E0003C0003C0003C000F800 03C001E00007C00FC000FFFFFF0000FFFFF8000025227EA129>I<00FFFFFFE000FFFFFF E000078003E000078000E0000F000060000F000060000F000040000F000040001E000040 001E004040001E004040001E004040003C008000003C008000003C038000003FFF800000 7FFF000000780300000078030000007803000000F002000000F002008000F002008000F0 00010001E000010001E000020001E000020001E000060003C0000C0003C0000C0003C000 380007C000F800FFFFFFF800FFFFFFF00023227EA125>I<00FFFFFFE000FFFFFFE00007 8003E000078000E0000F000060000F000060000F000040000F000040001E000040001E00 0040001E004040001E004040003C008000003C008000003C018000003C038000007FFF00 00007FFF00000078070000007803000000F002000000F002000000F002000000F0020000 01E000000001E000000001E000000001E000000003C000000003C000000003C000000007 C0000000FFFF000000FFFE00000023227EA120>I<00003F80200001FFE0400007E030C0 001F0009C0003C0007C000F800078001E000038003C000038007C00003800F800003000F 000001001F000001003E000003003E000002007C000000007C000000007C00000000F800 000000F800000000F800000000F800000000F8001FFF80F0001FFF80F000007800F00000 7800F80000F000F80000F000780000F000780000F0003C0001E0003C0001E0001E0003E0 000F8004E00007E038C00001FFF04000007F80000023247DA227>I<007FFC1FFF00FFFC 3FFF00078001E000078001E0000F0003C0000F0003C0000F0003C0000F0003C0001E0007 80001E000780001E000780001E000780003C000F00003C000F00003C000F00003FFFFF00 007FFFFE000078001E000078001E000078001E0000F0003C0000F0003C0000F0003C0000 F0003C0001E000780001E000780001E000780001E000780003C000F00003C000F00003C0 00F00007C001F0007FFC1FFF00FFFC3FFF0028227EA129>I<007FFC00FFFC0007800007 80000F00000F00000F00000F00001E00001E00001E00001E00003C00003C00003C00003C 0000780000780000780000780000F00000F00000F00000F00001E00001E00001E00001E0 0003C00003C00003C00007C0007FFC00FFFC0016227EA116>I<0003FFF00003FFF00000 0F0000000F0000001E0000001E0000001E0000001E0000003C0000003C0000003C000000 3C00000078000000780000007800000078000000F0000000F0000000F0000000F0000001 E0000001E0000001E0000001E0000003C0000003C0003803C0007803C000F8078000F807 8000F00F0000401E0000403C0000307000000FC000001C237DA11C>I<00FFFC01FF8000 FFFC01FF80000780007800000780006000000F00008000000F00010000000F0002000000 0F00040000001E00100000001E00200000001E00400000001E01000000003C0200000000 3C06000000003C0E000000003C2F00000000784F00000000788F00000000790780000000 7C0780000000F803C0000000F003C0000000F003E0000000F001E0000001E001E0000001 E000F0000001E000F0000001E00078000003C00078000003C00078000003C0003C000007 C0007E0000FFFC03FFC000FFFC03FFC00029227EA12A>I<00FFFE0000FFFE0000078000 00078000000F0000000F0000000F0000000F0000001E0000001E0000001E0000001E0000 003C0000003C0000003C0000003C00000078000000780000007800000078000000F00000 00F0000800F0000800F0001001E0001001E0001001E0002001E0002003C0006003C000C0 03C003C007C00FC0FFFFFF80FFFFFF801D227EA122>I<00FF800000FF8000FF800001FF 800007800002F0000007800002F0000009C00005E0000009C00005E0000009C00009E000 0009C00011E0000011C00013C0000011C00023C0000011C00043C0000010E00043C00000 20E0008780000020E0010780000020E0010780000020E0020780000040E0020F00000040 E0040F00000040E0080F0000004070080F0000008070101E0000008070201E0000008070 201E0000008070401E0000010070403C0000010070803C0000010039003C000001003900 3C000002003A0078000002003C0078000002003C007800000F003800F80000FFE0301FFF 8000FFE0301FFF800031227EA130>I<007F8003FF00FFC007FF0007C000700005E00060
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ndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmsl10 12 44 /Fo 44 122 df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
0000F00000F00000F00000F00000F00000F00001E00001E00001E00001E00001E00001E0 0003C00003C00003C00003C00003C00003C00007C000FFFC00FFFC0013227EA112>73 D<03FFF80003FFF800003E0000003C0000003C0000003C0000003C000000780000007800 000078000000780000007800000078000000F0000000F0000000F0000000F0000000F000 0000F0000001E0000001E0000001E0001001E0001001E0003001E0002003C0002003C000 6003C0006003C000C003C001C003C003C007C00FC0FFFFFF80FFFFFF801C227EA11F>76 D<03FE001FF803FE001FF8003F0003C0002F000100002F80010000278001000027800100 0043C002000043C002000041E002000041E002000040F002000040F00200008078040000 8078040000803C040000803C040000801E040000801E040001000F080001000F08000100 0F880001000788000100078800010003C800020003D000020001F000020001F000020000 F000020000F00006000070000F00006000FFE0002000FFE000200025227EA125>78 D<03FFFFC007FFFFF0003C00FC003C003E003C001E003C001F003C001F0078001F007800 1F0078001F0078001F0078001E0078003E00F0003C00F0007800F000F000F003E000FFFF 8000F0000001E0000001E0000001E0000001E0000001E0000001E0000003C0000003C000 0003C0000003C0000003C0000003C0000007C00000FFFC0000FFFC000020227EA122>80 D<0001FC00000F0780003C01E0007000F000E0007803C000380780003C0780001E0F0000 1E1E00001E1E00001F3E00001F3C00001F7C00001F7C00001FF800001FF800001FF80000 1FF800001FF800001FF000003EF000003EF000003EF000007CF000007CF0000078F80000 F8780000F0780F01E0381083C03C2043801E2047000E407E000720780001E0E008007FE0 080000600800007010000070300000787000007FE000003FE000003FC000001F8000000F 00202D7AA227>I<03FFFF800007FFFFF000003C00F800003C003C00003C001E00003C00 1F00003C001F000078001F000078001F000078001F000078001F000078003E000078003C 0000F000780000F000F00000F007C00000FFFE000000F00E000000F003000001E0038000 01E001C00001E001C00001E001E00001E001E00001E001E00003C003E00003C003E00003 C003E00003C003E00003C003E02003C003E04007C003E040FFFC01F080FFFC00F1800000 003E0023237EA125>I<1FFFFFFF3FFFFFFF3C01E01E3001E0062001E0026001E0024001 E0024003C0024003C0028003C0028003C0028003C0020003C00000078000000780000007 8000000780000007800000078000000F0000000F0000000F0000000F0000000F0000000F 0000001E0000001E0000001E0000001E0000001E0000001E0000003E00001FFFF8001FFF F80020227AA124>84 D<03FC000E07001F01801F01C01E00C00C00E00001C00001C0003F C003E1C00F01C01E01C0380380780380F00384F00384F00784F00784F00B887833D01FC1 E016157D9419>97 D<0780003F80003F00000700000700000700000700000700000E0000 0E00000E00000E00000E00000E00001C3F001CC1C01D00601E00701C00381C003838003C 38003C38003C38003C38003C38003C7000787000787000707000F07000E06801C0E80380 C60E0081F80016237BA21C>I<00FF000383800607C00C07C01C07803803007800007000 00F00000F00000F00000F00000F00000E00000E00000F000807001007001003806001C18 0007E00012157C9416>I<00000F00007F00007E00000E00000E00000E00000E00000E00 001C00001C00001C00001C00001C00001C007E3803C1380700B80E00781C003838003878 0070700070F00070F00070F00070F00070E000E0E000E0E000E0F000E07001E07003E038 05E01C19FC07E1FC18237CA21C>I<00FE000383800701C00C00E01C00E03800E07800E0 7000E0FFFFE0F00000F00000F00000F00000E00000E00000F00040700080300080380300 0E0C0003F00013157D9416>I<0003E0000E300018780030F80070F800E07000E00000E0 0001C00001C00001C00001C00001C00001C0007FFC007FFC000380000380000380000380 000700000700000700000700000700000700000E00000E00000E00000E00000E00000E00 001E0000FFE000FFE00015237EA20F>I<00000380001F84C00070E9C000E071C001C038 0003C0380003C03800078078000780780007807800078070000380F0000180E00003C380 00027E00000600000004000000060000000600000007FFC00003FFF00007FFF8001C007C 0030000C0060000C0060000C00C0000C00C0000C004000180060003000300060000C0380 0003FE00001A21809519>I<0078000003F8000003F00000007000000070000000700000 007000000070000000E0000000E0000000E0000000E0000000E0000000E0000001C1F800 01C60C0001D80E0001E0070001E0070001C0070003C00E0003800E0003800E0003800E00 03800E0003800E0007001C0007001C0007001C0007001C0007001C0007001C000F003C00 FFE3FF80FFE3FF8019237FA21C>I<006000F001F001F000E00000000000000000000000 000000000001C00FC00FC003C001C001C003800380038003800380038007000700070007 00070007000F007FE0FFC00C227FA10E>I<0000C00001E00003E00003E00001C0000000 000000000000000000000000000000000000000000000380003F80003F80000780000380 000380000700000700000700000700000700000700000E00000E00000E00000E00000E00 000E00001C00001C00001C00001C00001C00001C00003800703800F83800F87000F06000 61C0003F0000132C84A10F>I<0078000003F8000003F000000070000000700000007000
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ndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmbx10 12 74 /Fp 74 128 df<000FF83F00007FFDFFC001FC1FE3E003F03FC7E007E03FC7E00FC03F87 E00FC03F83C00FC01F80000FC01F80000FC01F80000FC01F80000FC01F80000FC01F8000 FFFFFFFC00FFFFFFFC000FC01F80000FC01F80000FC01F80000FC01F80000FC01F80000F C01F80000FC01F80000FC01F80000FC01F80000FC01F80000FC01F80000FC01F80000FC0 1F80000FC01F80000FC01F80000FC01F80000FC01F80000FC01F80007FF8FFF8007FF8FF F80023237FA221>11 D<0007F800007FFC0001FC0E0003F01F0007E03F000FC03F000FC0 3F000FC03F000FC01E000FC00C000FC000000FC000000FC0FF80FFFFFF80FFFFFF800FC0 1F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC0 1F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F807FF8 FFF07FF8FFF01C237FA220>I<0007F807F800007FFE7FFC0001FC0FFC0E0003F00FF01F 0007E01FE03F000FC03FC03F000FC03FC03F000FC01FC03F000FC01FC01E000FC00FC00C 000FC00FC000000FC00FC000000FC00FC0FF80FFFFFFFFFF80FFFFFFFFFF800FC00FC01F 800FC00FC01F800FC00FC01F800FC00FC01F800FC00FC01F800FC00FC01F800FC00FC01F 800FC00FC01F800FC00FC01F800FC00FC01F800FC00FC01F800FC00FC01F800FC00FC01F 800FC00FC01F800FC00FC01F800FC00FC01F800FC00FC01F800FC00FC01F807FF87FF8FF F07FF87FF8FFF02C237FA230>14 D<10007C007E00FE00FF007F803FC00FC003E000E00B 0A7BA31D>18 D<3C7EFFFFFFFFFF7E7E7E7E7E7E3C3C3C3C3C3C18181818000000003C7E FFFFFFFF7E3C08237CA211>33 D<3C01E07E03F0FF07F8FF07F8FF87FCFF87FC7F83FC3D
81EC01800C01800C0300180300180700380600300C00601C00E03801C020010016127EA2 1E>I<3C007E00FF00FF00FF80FF807F803D800180018003000300070006000C001C0038 00200009127CA210>39 D<00180030006000C001C00380078007000F001E001E003E003C 003C007C007C007C007800F800F800F800F800F800F800F800F800F800F800F800F800F8 0078007C007C007C003C003C003E001E001E000F0007000780038001C000C00060003000 180D317BA416>I<C0006000300018001C000E000F000700078003C003C003E001E001E0 01F001F001F000F000F800F800F800F800F800F800F800F800F800F800F800F800F800F0 01F001F001F001E001E003E003C003C0078007000F000E001C00180030006000C0000D31 7DA416>I<00600000F00000F00000F00040F020F060F0F861F0FC63F03F6FC00FFF0001 F80001F8000FFF003F6FC0FC63F0F861F0F060F040F02000F00000F00000F00000600014 167CA41D>I<FFFCFFFCFFFCFFFCFFFC0E057F8D13>45 D<3C7EFFFFFFFF7E3C08087C87 10>I<00FF0003FFC00FC3F01F00F83E007C3E007C7C003E7C003E7C003E7C003EFC003F FC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003F 7C003E7C003E7E007E3E007C3E007C1F00F80FC3F003FFC000FF0018207E9F1D>48 D<00380000780003F800FFF800FDF80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F8007FFFF07FFFF014207C9F1D>I<03FC000F FF803C0FE07007F07C03F8FE01F8FE01FCFE01FCFE01FC7C01FC3801FC0001FC0001F800 03F80003F00007E0000FC0000F80001E00003C0000780000E00C01C00C03800C03000C06 001C0FFFF81FFFF83FFFF87FFFF8FFFFF8FFFFF816207D9F1D>I<00FF0007FFC00F03F0 1E01F83F01F83F01FC3F81FC3F01FC1F01FC0C01F80003F80003F00007E0000FC001FF00 01FF000003E00001F80001FC0000FE0000FE0000FF7C00FF7C00FFFE00FFFE00FFFE00FE 7C01FC7801FC3C03F00FFFE001FF0018207E9F1D>I<0000700000F00001F00001F00003 F00007F0000FF0001DF00039F00031F00061F000C1F001C1F00381F00701F00601F00C01 F01801F03801F07001F0E001F0FFFFFFFFFFFF0003F00003F00003F00003F00003F00003 F00003F0007FFF007FFF18207E9F1D>I<1000301E01F01FFFE01FFFE01FFFC01FFF001F FE001FF80018000018000018000018000019FE001FFF801E07E01803F01001F00001F800 01F80001FC0001FC7801FCFC01FCFC01FCFC01FCFC01F8F801F86003F03003E01C0FC00F FF0003FC0016207D9F1D>I<001FC000FFF001F03807C0780F80FC1F00FC1F00FC3F00FC 7E00787E00007E0000FE0000FE3FC0FE7FF0FE80F8FF807CFF007EFF007EFE007FFE007F FE007FFE007F7E007F7E007F7E007F3E007E3E007E1F007C0F00F807C1F003FFE000FF00 18207E9F1D>I<6000007800007FFFFF7FFFFF7FFFFE7FFFFC7FFFF87FFFF0E00060E000 60C000C0C00180C00300000600000E00000C00001C0000380000380000780000780000F0 0000F00001F00001F00001F00003F00003F00003F00003F00003F00003F00001E00000C0 0018227DA11D>I<00FF0003FFE00701F00E00781C00781C003C3C003C3E003C3F003C3F C0383FF0781FF8F01FFFE00FFF8007FFE003FFF007FFF81F3FFC3C0FFE7807FE7801FFF0 007FF0001FF0001FF0000FF0000FF8000E78001C3C001C1F00F80FFFE001FF0018207E9F 1D>I<00FF0007FFC00F83E01F00F03E00F87E007C7E007CFE007EFE007EFE007EFE007F FE007FFE007FFE007F7E00FF7E00FF3E01FF1F017F0FFE7F03FC7F00007F00007E00007E 1E007E3F00FC3F00FC3F00F83F01F01E03E01C0FC00FFF0003FC0018207E9F1D>I<3C7E FFFFFFFF7E3C0000000000003C7EFFFFFFFF7E3C08167C9510>I<07FC001FFF803807C0 7003E0F803F0FC03F0FC03F0FC03F07803E00007E0000FC0001F00003E00003800007000 00700000600000E00000C00000C00000C00000C00000C000000000000000000000000000 01E00003F00007F80007F80007F80007F80003F00001E00014237DA21B>63 D<0000300000000078000000007800000000FC00000000FC00000000FC00000001FE0000 0001FE00000003FF000000037F000000037F000000063F800000063F8000000C3FC00000 0C1FC000001C1FE00000180FE00000180FE000003007F000003007F000007007F8000060 03F800007FFFF80000FFFFFC0000C001FC00018001FE00018000FE00038000FF00030000 7F000300007F000600003F800F00003F80FFE007FFFCFFE007FFFC26227EA12B>65 D<FFFFFF8000FFFFFFF00007E001FC0007E000FE0007E0007F0007E0007F0007E0007F80 07E0003F8007E0003F8007E0003F8007E0007F8007E0007F0007E0007F0007E000FE0007 E003FC0007FFFFF00007FFFFF00007E001FE0007E0007F0007E0003F8007E0003FC007E0 001FC007E0001FE007E0001FE007E0001FE007E0001FE007E0001FE007E0001FE007E000 3FC007E0003F8007E0007F8007E001FE00FFFFFFFC00FFFFFFE00023227EA129>I<0001 FF0040001FFFC1C0007F80F3C001FC001FC003F0000FC007E00007C00FC00003C01FC000 03C03F800001C03F800001C07F800000C07F000000C07F000000C0FF00000000FF000000 00FF00000000FF00000000FF00000000FF00000000FF00000000FF000000007F00000000 7F000000C07F800000C03F800000C03F800001C01FC00001800FC000018007E000030003 F000060001FC001C00007F807800001FFFE0000001FF000022227DA129>I<FFFFFF8000
FFFFFFF80007F001FC0007F0007F0007F0003F8007F0000FC007F0000FE007F00007E007 F00007F007F00003F007F00003F807F00003F807F00003F807F00003FC07F00003FC07F0 0003FC07F00003FC07F00003FC07F00003FC07F00003FC07F00003FC07F00003FC07F000 03F807F00003F807F00003F807F00007F007F00007F007F0000FE007F0000FC007F0001F 8007F0007F0007F001FE00FFFFFFF800FFFFFF800026227EA12C>I<FFFFFFFE00FFFFFF FE0007F000FE0007F0001E0007F0000E0007F0000E0007F000070007F000070007F00003 0007F00C030007F00C030007F00C030007F00C000007F01C000007F03C000007FFFC0000 07FFFC000007F03C000007F01C000007F00C000007F00C000007F00C018007F00C018007 F000018007F000030007F000030007F000030007F000070007F000070007F0000F0007F0 001F0007F000FE00FFFFFFFE00FFFFFFFE0021227EA126>I<FFFFFFFCFFFFFFFC07F000 FC07F0003C07F0001C07F0000C07F0000E07F0000E07F0000607F0000607F00C0607F00C 0607F00C0007F00C0007F01C0007F03C0007FFFC0007FFFC0007F03C0007F01C0007F00C 0007F00C0007F00C0007F00C0007F0000007F0000007F0000007F0000007F0000007F000 0007F0000007F00000FFFFE000FFFFE0001F227EA124>I<0001FF0020001FFFE0E0007F 8079E001FC001FE003F80007E007E00003E00FC00001E01FC00001E03F800000E03F8000 00E07F800000607F000000607F00000060FF00000000FF00000000FF00000000FF000000 00FF00000000FF00000000FF0007FFFEFF0007FFFE7F00000FE07F00000FE07F80000FE0 3F80000FE03F80000FE01FC0000FE00FE0000FE007E0000FE003F8000FE001FC001FE000 7F8073E0001FFFE1E00001FF806027227DA12D>I<FFFF81FFFFFFFF81FFFF07F0000FE0 07F0000FE007F0000FE007F0000FE007F0000FE007F0000FE007F0000FE007F0000FE007 F0000FE007F0000FE007F0000FE007F0000FE007F0000FE007FFFFFFE007FFFFFFE007F0 000FE007F0000FE007F0000FE007F0000FE007F0000FE007F0000FE007F0000FE007F000 0FE007F0000FE007F0000FE007F0000FE007F0000FE007F0000FE007F0000FE007F0000F E0FFFF81FFFFFFFF81FFFF28227EA12D>I<FFFF80FFFF8007F00007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F000FFFF80FFFF8011227EA116>I<FFFF800FFEFFFF800FFE07F00001E007F0 00018007F000070007F0000E0007F000180007F000300007F000600007F001C00007F003 800007F006000007F00C000007F03C000007F07E000007F0FE000007F1FF000007F33F80 0007FE3FC00007FC1FE00007F00FE00007F007F00007F003F80007F003FC0007F001FC00 07F000FE0007F0007F0007F0007F8007F0003FC007F0001FC007F0000FE007F0000FF0FF FF80FFFFFFFF80FFFF28227EA12D>75 D<FFFFC000FFFFC00007F0000007F0000007F000 0007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F000 0007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0001807F000 1807F0001807F0001807F0003807F0003807F0007007F0007007F000F007F001F007F007 F0FFFFFFF0FFFFFFF01D227EA122>I<FFF0000007FF80FFF800000FFF8007F800000FF0 0006FC00001BF00006FC00001BF000067E000033F000067E000033F000067E000033F000 063F000063F000063F000063F000061F8000C3F000061F8000C3F000060FC00183F00006 0FC00183F0000607E00303F0000607E00303F0000603F00603F0000603F00603F0000603 F00603F0000601F80C03F0000601F80C03F0000600FC1803F0000600FC1803F00006007E 3003F00006007E3003F00006003F6003F00006003F6003F00006001FC003F00006001FC0 03F00006001FC003F00006000F8003F00006000F8003F000FFF007007FFF80FFF007007F FF8031227EA136>I<FFF0000FFFFFF8000FFF07FC00006007FE00006006FF000060067F 000060063F800060061FC00060060FE00060060FF000600607F800600603FC00600601FE 00600600FE006006007F006006003F806006001FC06006001FE06006000FF060060007F8 60060003FC60060001FC60060000FE600600007F600600003FE00600003FE00600001FE0 0600000FE006000007E006000003E006000001E006000000E0FFF0000060FFF000006028 227EA12D>I<0007FE0000003FFFC00000FE07F00003F801FC0007F000FE000FE0007F00 1FC0003F801F80001F803F80001FC03F80001FC07F00000FE07F00000FE07F00000FE0FF 00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00 000FF0FF00000FF07F00000FE07F80001FE07F80001FE03F80001FC01FC0003F801FC000 3F800FE0007F0007F000FE0003F801FC0000FE07F000003FFFC0000007FE000024227DA1 2B>I<FFFFFF8000FFFFFFF00007F003F80007F001FC0007F000FE0007F0007F0007F000 7F0007F0007F8007F0007F8007F0007F8007F0007F8007F0007F8007F0007F0007F0007F 0007F000FE0007F001FC0007F003F80007FFFFF00007FFFF800007F000000007F0000000 07F000000007F000000007F000000007F000000007F000000007F000000007F000000007 F000000007F000000007F000000007F0000000FFFF800000FFFF80000021227EA127>I< 0007FE0000003FFFC00000FE07F00003F801FC0007F000FE000FE0007F001FC0003F801F C0003F803F80001FC03F80001FC07F80001FE07F00000FE07F00000FE0FF00000FF0FF00
000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF0000 0FF07F00000FE07F80001FE07F80001FE03F80001FC01F80F01F801FC3F83F800FE70C7F 0007F606FE0003FE03FC0000FF07F000003FFFC0000007FFC030000001E030000001F070 000001FFF0000001FFF0000001FFE0000000FFE0000000FFC00000007FC00000003F8000 00001E00242C7DA12B>I<FFFFFF0000FFFFFFE00007F007F80007F001FC0007F000FE00 07F0007F0007F0007F8007F0007F8007F0007F8007F0007F8007F0007F8007F0007F8007 F0007F0007F000FE0007F001FC0007F007F80007FFFFE00007FFFF800007F00FC00007F0 03F00007F003F80007F001F80007F001FC0007F001FC0007F001FC0007F001FC0007F001 FE0007F001FE0007F001FE0307F001FF0307F000FF0307F0007F86FFFF803FFCFFFF800F F828227EA12B>I<01FE020007FFCE001F01FE003C007E003C001E0078000E0078000E00 F8000600F8000600FC000600FC000000FF000000FFF000007FFF80003FFFE0003FFFF800 1FFFFC0007FFFE0003FFFE00003FFF000001FF0000003F8000001F8000001F80C0000F80 C0000F80C0000F80E0000F00E0000F00F0001E00FC001C00FF807800E7FFF000807FC000 19227DA120>I<7FFFFFFFC07FFFFFFFC07E03F80FC07803F803C07003F801C06003F800 C0E003F800E0E003F800E0C003F80060C003F80060C003F80060C003F800600003F80000 0003F800000003F800000003F800000003F800000003F800000003F800000003F8000000 03F800000003F800000003F800000003F800000003F800000003F800000003F800000003 F800000003F800000003F800000003F8000001FFFFF00001FFFFF00023217EA028>I<FF FF801FFEFFFF801FFE07F00000C007F00000C007F00000C007F00000C007F00000C007F0 0000C007F00000C007F00000C007F00000C007F00000C007F00000C007F00000C007F000 00C007F00000C007F00000C007F00000C007F00000C007F00000C007F00000C007F00000 C007F00000C007F00000C007F00000C007F00000C003F000018003F800018001F8000300 00FC000600007E000C00003F807800000FFFF0000000FF800027227EA12C>I<FFFF8007 FFFFFF8007FF07F00000E007F800006003F80000C003FC0000C001FC00018001FC000180 01FE00038000FE00030000FF000700007F000600007F800600003F800C00003FC00C0000 1FC01800001FC01800000FE03000000FE03000000FF070000007F060000007F860000003 F8C0000003FCC0000001FD80000001FD80000000FF00000000FF00000000FF000000007E 000000007E000000003C000000003C0000000018000028227FA12B>I<FFFF07FFF80FFF FFFF07FFF80FFF0FF0007F80007007F0003F80006007F0003F80006003F8003FC000C003 F8001FC000C003FC001FC000C001FC003FE0018001FC003FE0018001FE007FF0038000FE 0067F0030000FE0067F00300007F00C3F80600007F00C3F80600007F80C3FC0E00003F81 81FC0C00003F8181FC0C00001FC381FE1800001FC300FE1800001FE300FE1800000FE600 7F3000000FE6007F3000000FFE007FF0000007FC003FE0000007FC003FE0000003F8001F C0000003F8001FC0000003F8001FC0000001F0000F80000001F0000F80000000F0000F00 000000E0000700000000E00007000038227FA13B>I<3FFFFFF03FFFFFF03FC00FE03E00 1FE03C001FC038003F8070007F8070007F007000FE006001FE006001FC006003F8000007 F8000007F000000FE000001FE000001FC000003F8000007F8000007F000000FE003001FE 003001FC003003F8003007F8007007F000700FE000701FE000E01FC000E03F8001E07F80 03E07F001FE0FFFFFFE0FFFFFFE01C227DA123>90 D<0200100E00701C00E01800C03001 80700380600300600300C00600C00600DE06F0FF07F8FF87FCFF87FC7F83FC7F83FC3F01 F81E00F016127BA21E>92 D<07FE00001FFF80003F07E0003F03F0003F01F0003F01F800 1E01F8000001F8000001F800003FF80003FDF8001F81F8003E01F8007C01F800F801F800 F801F800F801F800F801F8007C02F8007E0CF8001FF87F8007E03F8019167E951C>97 D<FF800000FF8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000 001F8000001F8000001F8000001F8000001F87F0001FBFFC001FF03E001FC01F001F800F 801F800FC01F8007C01F8007E01F8007E01F8007E01F8007E01F8007E01F8007E01F8007 E01F8007C01F8007C01F800FC01F800F801FC01F001E707E001C3FFC00180FE0001B237E A220>I<00FF8007FFE00F83F01F03F03E03F07E03F07C01E07C0000FC0000FC0000FC00 00FC0000FC0000FC00007C00007E00007E00003E00181F00300FC06007FFC000FF001516 7E9519>I<0001FF000001FF0000003F0000003F0000003F0000003F0000003F0000003F 0000003F0000003F0000003F0000003F0000003F0000FE3F0007FFBF000FC1FF001F007F 003E003F007E003F007C003F007C003F00FC003F00FC003F00FC003F00FC003F00FC003F 00FC003F00FC003F007C003F007E003F003E003F001F007F000F81FF0007FF3FE001FC3F E01B237EA220>I<00FE0007FF800F83C01E01E03E00F07E00F07C00F87C0078FC0078FF FFF8FFFFF8FC0000FC0000FC00007C00007C00003E00183E00181F00300F80E003FFC000 FF0015167E951A>I<001F8000FFE001F1F003E3F007E3F00FC3F00FC1E00FC0000FC000 0FC0000FC0000FC0000FC000FFFE00FFFE000FC0000FC0000FC0000FC0000FC0000FC000 0FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC000 7FFC007FFC0014237EA212>I<00FE0F8003FF9FC00F83E3C01F01F3C01E00F0003E00F8
003E00F8003E00F8003E00F8003E00F8001E00F0001F01F0000F83E0000BFF800008FE00 0018000000180000001C0000001FFFE0001FFFFC000FFFFF0007FFFF001FFFFF807C001F C078000FC0F80007C0F80007C0F80007C07C000F803E001F001F807E000FFFFC0001FFE0 001A217F951D>I<FF800000FF8000001F8000001F8000001F8000001F8000001F800000 1F8000001F8000001F8000001F8000001F8000001F8000001F83F0001F8FFC001F987E00 1FA03E001FC03F001FC03F001F803F001F803F001F803F001F803F001F803F001F803F00 1F803F001F803F001F803F001F803F001F803F001F803F001F803F001F803F00FFF1FFE0 FFF1FFE01B237DA220>I<1E003F007F807F807F807F803F001E00000000000000000000 000000FF80FF801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F 801F801F801F80FFF0FFF00C247EA30F>I<003C007E00FF00FF00FF00FF007E003C0000 0000000000000000000003FF03FF003F003F003F003F003F003F003F003F003F003F003F 003F003F003F003F003F003F003F003F003F003F003F003F783FFC3FFC3EFC7C78783FF0 1FC0102E83A312>I<FF800000FF8000001F8000001F8000001F8000001F8000001F8000 001F8000001F8000001F8000001F8000001F8000001F8000001F80FF801F80FF801F803C 001F8030001F80E0001F81C0001F8300001F8600001F9E00001FBE00001FFF00001FDF80 001F8FC0001F07C0001F07E0001F03F0001F01F8001F00F8001F00FC001F007E00FFE1FF C0FFE1FFC01A237EA21E>I<FF80FF801F801F801F801F801F801F801F801F801F801F80 1F801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F80 1F801F801F80FFF0FFF00C237EA20F>I<FF03F803F800FF0FFE0FFE001F183F183F001F 201F201F001F401FC01F801F401FC01F801F801F801F801F801F801F801F801F801F801F 801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F 801F801F801F801F801F801F801F801F801F801F801F801F801F801F80FFF0FFF0FFF0FF F0FFF0FFF02C167D9531>I<FF03F000FF0FFC001F187E001F203E001F403F001F403F00 1F803F001F803F001F803F001F803F001F803F001F803F001F803F001F803F001F803F00 1F803F001F803F001F803F001F803F001F803F00FFF1FFE0FFF1FFE01B167D9520>I<00 FF0007FFE00F81F01F00F83E007C7C003E7C003E7C003EFC003FFC003FFC003FFC003FFC 003FFC003FFC003F7C003E7E007E3E007C1F00F80F81F007FFE000FF0018167E951D>I< FF87F000FFBFFC001FF07E001FC01F001F800F801F800FC01F800FC01F8007E01F8007E0 1F8007E01F8007E01F8007E01F8007E01F8007E01F8007C01F800FC01F800FC01F801F80 1FC01F001FF07E001FBFFC001F8FE0001F8000001F8000001F8000001F8000001F800000 1F8000001F8000001F800000FFF00000FFF000001B207E9520>I<00FE030007FF07000F C1CF001F00DF003F007F007E003F007E003F007C003F00FC003F00FC003F00FC003F00FC 003F00FC003F00FC003F00FC003F007E003F007E003F003E007F001F00FF000FC1FF0007 FF3F0000FC3F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000 003F000001FFE00001FFE01B207E951E>I<FF0F80FF1FE01F33F01F63F01F43F01F43F0 1FC1E01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8000 1F80001F8000FFF800FFF80014167E9518>I<07F9801FFF80380780700380F00180F001 80F80000FF0000FFF8007FFE003FFF001FFF8007FF80003FC0C007C0C003C0E003C0E003 C0F00380FC0F00EFFE00C3F80012167E9517>I<01800001800001800001800003800003 80000780000780000F80003F8000FFFF00FFFF001F80001F80001F80001F80001F80001F 80001F80001F80001F80001F80001F80001F81801F81801F81801F81801F81800F83000F C30007FE0001F80011207F9F16>I<FF81FF00FF81FF001F803F001F803F001F803F001F 803F001F803F001F803F001F803F001F803F001F803F001F803F001F803F001F803F001F 803F001F803F001F803F001F807F001F80FF000FC1BF0007FF3FE001FC3FE01B167D9520 >I<FFF01FE0FFF01FE01FC007000FC006000FE00E0007E00C0007F00C0003F0180003F0 180001F8300001F8300000FC600000FC600000FEE000007EC000007FC000003F8000003F 8000001F0000001F0000000E0000000E00001B167F951E>I<FFF3FF83FCFFF3FF83FC1F 807C00E00FC07C00C00FC03E00C00FE03E01C007E07F018007E07F018003F07F030003F0 CF830003F8CF870001F9CFC60001F987C60000FD87EC0000FF03EC0000FF03FC00007F03 F800007E01F800003E01F000003C00F000003C00F000001800600026167F9529>I<FFF0 7FC0FFF07FC00FC01C0007E0380007F0700003F0E00001F8C00000FD8000007F8000007F 0000003F0000001F8000003FC0000037E0000067F00000C3F00001C1F8000380FC000700 FE000E007F00FF81FFE0FF81FFE01B167F951E>I<FFF01FE0FFF01FE01FC007000FC006 000FE00E0007E00C0007F00C0003F0180003F0180001F8300001F8300000FC600000FC60 0000FEE000007EC000007FC000003F8000003F8000001F0000001F0000000E0000000E00 00000C0000000C00000018000078180000FC380000FC300000FC60000069C000007F8000 001F0000001B207F951E>I<3807007C0F80FE1FC0FE1FC0FE1FC07C0F8038070012077B A31D>127 D E %EndDVIPSBitmapFont
%DVIPSBitmapFont: Fq cmti10 12 49 /Fq 49 128 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr10 12 98 /Fr 98 128 df<FFFFFF80FFFFFF8007800F80078003800780018007800080078000C007 8000C0078000400780004007800040078000400780000007800000078000000780000007 800000078000000780000007800000078000000780000007800000078000000780000007 800000078000000780000007800000078000000780000007800000FFFE0000FFFE00001A 227EA11F>0 D<0000200000000070000000007000000000F800000000F8000000017C00 0000013C000000023E000000021E000000041F000000040F000000080F80000008078000 001007C000001003C000002003E000002001E000004000F00000C000F800008000780001 80007C000100003C000300003E000200001E000600001F000400000F000C00000F800800 00078010000007C010000003C020000003E020000001E07FFFFFFFF07FFFFFFFF0FFFFFF FFF825237EA22A>I<FFFFFFF0FFFFFFF0780003F07C0000703E0000301E0000301F0000 180F80001807C0000803C0000803E0000801F0000800F0000000F80000007C0000003C00 00003E0000001E0000000C00000008000000100000002000000020000800400008008000 08010000080200001804000018080000301000003010000070200003F07FFFFFF0FFFFFF F01D227DA124>6 D<001FF00000783C0001C0070003800380070001C00F0001E01E0000 F03E0000F83C0000787C00007C7C00007C7C00007C7C00007C7C00007C7C00007C7C0000 7C3C0000783E0000F81E0000F01E0000F00F0001E00F0001E0070001C003800380038003 800180030001C0070080C0060280C006024040040440600C04402008047FE00FFC3FE00F F83FE00FF81F237EA224>10 D<000FC0F80078338600E03F0F01807E1F03807E1F07007C 0E07003C0007001C0007001C0007001C0007001C0007001C0007001C0007001C00FFFFFF
E0FFFFFFE007001C0007001C0007001C0007001C0007001C0007001C0007001C0007001C 0007001C0007001C0007001C0007001C0007001C0007001C0007001C0007001C0007001C 007FF1FFE07FF1FFE0202380A21D>I<000FC0000078300000E0080001803C0003807C00 07007C0007007C0007003800070000000700000007000000070000000700000007000000 FFFFFC00FFFFFC0007003C0007001C0007001C0007001C0007001C0007001C0007001C00 07001C0007001C0007001C0007001C0007001C0007001C0007001C0007001C0007001C00 07001C007FF1FFC07FF1FFC01A2380A21C>I<000FEC0000701C0000E07C0001807C0003 807C0007003C0007001C0007001C0007001C0007001C0007001C0007001C0007001C0007 001C00FFFFFC00FFFFFC0007001C0007001C0007001C0007001C0007001C0007001C0007 001C0007001C0007001C0007001C0007001C0007001C0007001C0007001C0007001C0007 001C0007001C007FF1FFC07FF1FFC01A2380A21C>I<000FC03F00007031E0C000E00B80 2001803E00F003807E01F007007C01F007007C01F007003C00E007001C000007001C0000 07001C000007001C000007001C000007001C0000FFFFFFFFF0FFFFFFFFF007001C00F007 001C007007001C007007001C007007001C007007001C007007001C007007001C00700700 1C007007001C007007001C007007001C007007001C007007001C007007001C007007001C 007007001C00707FF1FFC7FF7FF1FFC7FF282380A22A>I<0380038007800E001C001800 30006000C0008000090A76A219>19 D<FFFFC012017D9C19>22 D<70F8F8F8F8F8F8F870 7070707070707070707070202020202020000000000070F8F8F87005247CA30E>33 D<7038F87CFC7EFC7E743A04020402040204020804080410081008201040200F0F7EA219 >I<0000C00C000000C00C000000C00C0000018018000001801800000180180000018018 000003003000000300300000030030000003003000000300300000060060000006006000 00060060000006006000000C00C000FFFFFFFFE0FFFFFFFFE00018018000001801800000 1801800000180180000030030000003003000000300300000030030000FFFFFFFFE0FFFF FFFFE0006006000000C00C000000C00C000000C00C000000C00C00000180180000018018 000001801800000180180000030030000003003000000300300000030030000006006000 0006006000000600600000232D7DA22A>I<003C000000006200000000C1000000018080 000001808000000380800000038080000003808000000380800000038100000003810000 000382000000038400000001C800000001C800000001D001FFC000E001FFC000E0003C00 00E0001800017000100002700020000438002000083C004000181C008000300E00800030 0F0100007007020000F003820000F003C40000F001C80000F000F00080F8007000807800 7801807C019C03003E070F06000FFE07FE0003F001F80022257EA327>38 D<70F8FCFC7404040404080810102040060F7CA20E>I<00200040008001000300060004 000C000C00180018003000300030007000600060006000E000E000E000E000E000E000E0 00E000E000E000E000E000E000E0006000600060007000300030003000180018000C000C 0004000600030001000080004000200B327CA413>I<800040002000100018000C000400 060006000300030001800180018001C000C000C000C000E000E000E000E000E000E000E0 00E000E000E000E000E000E000E000C000C000C001C00180018001800300030006000600 04000C00180010002000400080000B327DA413>I<00C00000C00000C00000C00000C000 E0C1C0F0C3C038C7000EDC0003F00000C00003F0000EDC0038C700F0C3C0E0C1C000C000 00C00000C00000C00000C00012157DA419>I<0001800000018000000180000001800000 018000000180000001800000018000000180000001800000018000000180000001800000 0180000001800000018000FFFFFFFFFFFFFFFF0001800000018000000180000001800000 018000000180000001800000018000000180000001800000018000000180000001800000 018000000180000001800020227D9C27>I<70F8FCFC7404040404080810102040060F7C 840E>I<FFF0FFF00C027F8B11>I<70F8F8F87005057C840E>I<0000400000C000018000 0180000180000300000300000300000600000600000600000C00000C00000C0000180000 180000180000300000300000300000600000600000600000C00000C00000C00001800001 80000180000300000300000300000600000600000600000C00000C00000C000018000018 0000180000300000300000300000600000600000600000C00000C0000012317DA419>I< 01F800070E000E07001C03803801C03801C07801E07000E07000E07000E0F000F0F000F0 F000F0F000F0F000F0F000F0F000F0F000F0F000F0F000F0F000F0F000F0F000F0F000F0 7000E07000E07000E07801E03801C03801C01C03800E0700070E0001F80014227EA019> I<008003800F80FF80F38003800380038003800380038003800380038003800380038003 800380038003800380038003800380038003800380038003800380FFFEFFFE0F217BA019 >I<03F8000FFE00181F803007C04003C04001E0F801E0FC01F0FC00F0FC00F07800F030 01F00001E00001E00003E00003C0000780000700000E00001C0000380000700000E00000 80000100000200100400100800101000302000207FFFE0FFFFE0FFFFE014217EA019>I< 01F8000FFE001C0F803003C03803C07C03E07C01E07C01E03C03E01803E00003C0000380 000780000E00001C0003F800000E000007800003C00003E00001E00001F00001F03001F0
7801F0FC01F0FC01F0FC01E0F803E04003C02007801C0F000FFE0003F80014227EA019> I<000180000180000380000780000780000B800013800033800023800043800083800083 80010380020380060380040380080380100380100380200380400380C00380FFFFFCFFFF FC000380000380000380000380000380000380000380007FFC007FFC16217FA019>I<10 00401E03801FFF001FFE001FFC0013F00010000010000010000010000010000010000010 F8001306001403001801801001C00001E00000E00000F00000F00000F07000F0F800F0F8 00F0F800F0F800E0C001E04001C06003C03003801C0F000FFE0003F00014227EA019>I< 003E0000FF8003C0C00700E00E01E01C03E01C03E03801C0380000780000700000700000 F0F800F30600F40300F40180F801C0F800E0F000E0F000F0F000F0F000F0F000F0F000F0 7000F07000F07800E03800E03801E01C01C01C03800F070003FE0001F80014227EA019> I<4000006000007FFFF07FFFF07FFFE0400020C000408000808001008001000002000004 0000040000080000100000100000300000600000600000E00000E00001C00001C00001C0 0003C00003C00003C00003C00007C00007C00007C00007C00007C00007C0000380001423 7DA119>I<01F80007FE000E07801801C03000C06000606000606000606000607000607C 00C03E00801F81000FC60007FC0003F80001FE00067F000C1F80100FC03003E06001E060 0070C00070C00030C00030C00030C000306000607000403800C01E07800FFE0001F80014 227EA019>I<01F80007FE000E0F001C03803803807801C07001C07001E0F000E0F000E0 F000F0F000F0F000F0F000F0F000F07000F07001F03801F01802F00C02F0060CF001F0F0 0000E00000E00001E00001C03801C07C03807C0380780700300E00383C001FF80007E000 14227EA019>I<70F8F8F870000000000000000000000070F8F8F87005157C940E>I<70F8 F8F870000000000000000000000070F8F8F87808080808101010204040051F7C940E>I< FFFFFFFFFFFFFFFF00000000000000000000000000000000000000000000000000000000 00000000FFFFFFFFFFFFFFFF200C7D9127>61 D<07F000181C00200E00400700F00780F8 0780F80780F80780700780000F00000E00001C0000380000700000600000C00000800000 800001800001000001000001000001000001000001000000000000000000000000000000 000003800007C00007C00007C00003800011237DA218>63 D<000FF00000300C0000C003 00010000800600006004000020080000101007E008201C100420300C044060020240C003 C241C001C281C001C1838001C1838001C1838001C1838001C1838001C1838001C1838001 C181C001C141C001C140C003C1406003C220300DC2201C10C41007E07808000000040000 00060000000100000F00C0003C003003E0000FFC0020237DA227>I<0001800000018000 000180000003C0000003C0000003C0000005E0000005E000000DF0000008F0000008F000 0010F800001078000010780000203C0000203C0000203C0000401E0000401E0000401E00 00800F0000800F0000FFFF0001FFFF800100078001000780020003C0020003C0060003E0 040001E0040001E00C0001F01E0001F0FF800FFFFF800FFF20237EA225>I<FFFFF800FF FFFE0007800F80078007C0078003E0078001E0078001F0078001F0078001F0078001F007 8001F0078001E0078003E0078007C007800F8007803E0007FFFE0007800780078003C007 8001E0078001F0078000F0078000F8078000F8078000F8078000F8078000F8078000F807 8001F0078003F0078003E007800FC0FFFFFF00FFFFFC001D227EA123>I<0007F008003F FC1800FC061801F0013803C000F8078000780F0000381E0000381E0000183C0000183C00 00187C0000087C00000878000008F8000000F8000000F8000000F8000000F8000000F800 0000F8000000F8000000780000007C0000087C0000083C0000083C0000081E0000101E00 00100F0000200780002003C0004001F0018000FC0700003FFC000007F0001D247DA224> I<FFFFF800FFFFFE0007800F80078003C0078001E0078000F00780007807800078078000 3C0780003C0780003E0780001E0780001E0780001F0780001F0780001F0780001F078000 1F0780001F0780001F0780001F0780001F0780001E0780001E0780003E0780003C078000 3C07800078078000F0078001E0078003C007800F80FFFFFF00FFFFF80020227EA126>I< FFFFFFE0FFFFFFE0078003E0078000E00780006007800020078000300780003007800010 07801010078010100780101007801000078030000780700007FFF00007FFF00007807000 078030000780100007801000078010040780100407800004078000080780000807800008 07800018078000180780003807800078078001F0FFFFFFF0FFFFFFF01E227EA122>I<FF FFFFE0FFFFFFE0078003E0078000E0078000600780002007800030078000300780001007 80001007801010078010100780100007801000078030000780700007FFF00007FFF00007 807000078030000780100007801000078010000780100007800000078000000780000007 80000007800000078000000780000007800000FFFE0000FFFE00001C227EA121>I<0007 F80400001FFE0C00007E070C0001F0009C0003C0007C000780003C000F00001C001F0000 1C001E00000C003E00000C003C00000C007C000004007C000004007800000400F8000000 00F800000000F800000000F800000000F800000000F800000000F800000000F8000FFF80 78000FFF807C00003C007C00003C003C00003C003E00003C001E00003C001F00003C000F 00003C000780003C0003E0005C0001F000DC00007E038C00001FFE04000007F800002124
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ndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmr12 17.28 34 /Fs 34 123 df<387CFEFEFF7D39010101010202020404080810204008157A8614>44 D<387CFEFEFE7C3807077A8614>46 D<000C00001C00003C0001FC003FFC00FE7C00C07C 00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C 00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C 00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C 00007C0000FE007FFFFE7FFFFE172F7AAE23>49 D<003F800001FFF0000781FC000E003E 0018001F0038000F80300007C0600007E0600007E0F80003F0FC0003F0FE0003F0FE0003 F0FE0003F0FE0003F07C0003F0000003F0000003E0000007E0000007C000000FC000000F 8000001F0000001F0000003E0000003C00000078000000F0000001E0000003C000000780 0000060000000C000000180000003000000060003000C000300180003003000060060000 600C000060080000E01FFFFFE03FFFFFE07FFFFFC0FFFFFFC0FFFFFFC01C2F7DAE23>I< 0000060000000E0000000E0000001E0000003E0000003E0000007E000000FE000000BE00 0001BE0000033E0000023E0000063E00000C3E0000083E0000183E0000303E0000203E00 00603E0000C03E0000803E0001803E0003003E0002003E0006003E000C003E0008003E00 18003E0030003E0020003E0060003E00C0003E00FFFFFFFCFFFFFFFC00003E0000003E00 00003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000007F00 001FFFFC001FFFFC1E2F7EAE23>52 D<0C0001800F800F000FFFFE000FFFFC000FFFF800 0FFFF0000FFF80000C3800000C0000000C0000000C0000000C0000000C0000000C000000 0C0000000C0000000C0000000C0000000C1FC0000C6078000D801C000E000E000C000F00 08000780000007C0000007C0000003E0000003E0000003E0000003F0000003F0000003F0 380003F07C0003F0FC0003F0FC0003F0FC0003F0F80003E0C00003E0600007C0600007C0
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
8001F30003F000018000FE0003F000018000FE0003F000018000FE0003F0000180007C00 03F00003C0007C0003F0000FF000380007F800FFFF003801FFFFE0FFFF003801FFFFE03B 317EB040>I<FFFFFFF80000FFFFFFFF000003F8001FC00001F80003F00001F80001F800 01F80000FC0001F800007E0001F800007E0001F800003F0001F800003F0001F800003F80 01F800003F8001F800003F8001F800003F8001F800003F8001F800003F8001F800003F00 01F800003F0001F800007E0001F800007E0001F80000FC0001F80001F80001F80003E000 01F8001FC00001FFFFFE000001F80000000001F80000000001F80000000001F800000000 01F80000000001F80000000001F80000000001F80000000001F80000000001F800000000 01F80000000001F80000000001F80000000001F80000000001F80000000001F800000000 01F80000000001F80000000001F80000000001F80000000001F80000000003FC00000000 FFFFF0000000FFFFF000000029317EB030>80 D<003F801001FFF03003C078700F000E70 0E0007F01C0003F0380001F0780000F0700000F070000070F0000070F0000070F0000030 F0000030F8000030F8000030FC0000007C0000007E0000007F8000003FF000001FFF0000 0FFFF00007FFFE0003FFFF0001FFFF80003FFFC00003FFE000007FF0000007F0000003F8 000001F8000000F80000007C0000007CC000003CC000003CC000003CC000003CC000003C E0000038E0000038F0000078F0000070F8000070FC0000E0FE0001C0E3800380E1F00F00 C07FFE00800FF0001E337CB127>83 D<FFFFC01FFFF001FFF8FFFFC01FFFF001FFF807F8 0001FF00003FC003F00000FE00001F0003F000007E00000E0003F000007E00000E0001F8 00007E00000C0001F800007F00000C0001F800007F00000C0000FC00007F0000180000FC 0000DF8000180000FC0000DF80001800007E0000DF80003000007E0001DFC0003000007E 00018FC0003000007F00018FC0007000003F00018FC0006000003F000307E0006000003F 000307E0006000001F800307E000C000001F800603F000C000001F800603F000C000000F C00603F0018000000FC00C01F8018000000FC00C01F80180000007E00C01F80300000007 E01C01FC0300000007E01800FC0300000007F01800FC0700000003F03800FE0600000003 F030007E0600000003F830007E0E00000001F830007E0C00000001F860003F0C00000001 F860003F0C00000000FC60003F1800000000FCC0001F9800000000FCC0001F9800000000 7EC0001FB0000000007F80000FF0000000007F80000FF0000000003F80000FE000000000 3F80000FE0000000003F000007E0000000003F000007E0000000001F000007C000000000 1E000003C0000000001E000003C0000000000E00000380000000000C0000018000004532 7FB048>87 D<00FF00000381E0000C00F000180078003E003C003F003E003F001E003F00 1F001E001F000C001F0000001F0000001F0000001F00000FFF0000FC1F0003E01F000780 1F001F001F003E001F003C001F007C001F00F8001F06F8001F06F8001F06F8001F06F800 3F0678003F067C004F063E008F8C0F8307F801FC01F01F1F7D9E23>97 D<07C0000000FFC0000000FFC00000000FC000000007C000000007C000000007C0000000 07C000000007C000000007C000000007C000000007C000000007C000000007C000000007 C000000007C000000007C000000007C000000007C000000007C1FC000007C703800007CC 01C00007F000700007E000780007E0003C0007C0001C0007C0001E0007C0001E0007C000 1F0007C0000F0007C0000F8007C0000F8007C0000F8007C0000F8007C0000F8007C0000F 8007C0000F8007C0000F8007C0000F0007C0000F0007C0001F0007C0001E0007C0001E00 07C0003C0007E000380007E000700007B000E000071801C00007060700000601FC000021 327EB127>I<003FC00000E0700003C00C00070006000F001F001E003F001E003F003C00 3F003C001E007C000C0078000000F8000000F8000000F8000000F8000000F8000000F800 0000F8000000F8000000F8000000780000007C0000003C0000003C0001801E0001801E00 03000F0003000780060003C01C0000E07000003FC000191F7D9E1F>I<000001F0000000 3FF00000003FF000000003F000000001F000000001F000000001F000000001F000000001 F000000001F000000001F000000001F000000001F000000001F000000001F000000001F0 00000001F000000001F000000001F000001FC1F000007031F00001C00DF000038007F000 070003F0000E0001F0001E0001F0003C0001F0003C0001F0007C0001F000780001F00078 0001F000F80001F000F80001F000F80001F000F80001F000F80001F000F80001F000F800 01F000F80001F000780001F0007C0001F0003C0001F0003C0001F0001C0001F0001E0003 F0000F0003F000070005F00001C019F80000E071FF80001F81FF8021327DB127>I<001F C00000E0700001803C0007001E000F000E001E0007001C0007803C0007803C0003807C00 03C0780003C0F80003C0F80003C0FFFFFFC0F8000000F8000000F8000000F8000000F800 000078000000780000007C0000003C0000003C0000C01E0000C00E000180070001000380 030001C00C0000703800001FC0001A1F7E9E1F>I<0003F0001E1800383C00787E00F07E 01F07E01E03C03E01803E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E000FFFFC0FFFFC003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00007F0007FFF807FFF8017327FB115>I<07C0
000000FFC0000000FFC00000000FC000000007C000000007C000000007C000000007C000 000007C000000007C000000007C000000007C000000007C000000007C000000007C00000 0007C000000007C000000007C000000007C000000007C07F000007C183C00007C601E000 07C801F00007D000F00007D000F80007E000F80007E000F80007C000F80007C000F80007 C000F80007C000F80007C000F80007C000F80007C000F80007C000F80007C000F80007C0 00F80007C000F80007C000F80007C000F80007C000F80007C000F80007C000F80007C000 F80007C000F80007C000F80007C000F8000FE001FC00FFFE1FFFC0FFFE1FFFC022327EB1 27>104 D<07000F801FC01FC01FC00F8007000000000000000000000000000000000000 000000000007C0FFC0FFC00FC007C007C007C007C007C007C007C007C007C007C007C007 C007C007C007C007C007C007C007C007C007C007C007C007C00FE0FFFEFFFE0F317EB014 >I<07C0FFC0FFC00FC007C007C007C007C007C007C007C007C007C007C007C007C007C0 07C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C0 07C007C007C007C007C007C007C007C007C007C007C007C00FE0FFFEFFFE0F327EB114> 108 D<07C07F0007F000FFC183C0183C00FFC600F0600F000FC800F8800F8007C8007880 078007D0007D0007C007E0007E0007C007E0007E0007C007C0007C0007C007C0007C0007 C007C0007C0007C007C0007C0007C007C0007C0007C007C0007C0007C007C0007C0007C0 07C0007C0007C007C0007C0007C007C0007C0007C007C0007C0007C007C0007C0007C007 C0007C0007C007C0007C0007C007C0007C0007C007C0007C0007C007C0007C0007C007C0 007C0007C007C0007C0007C007C0007C0007C00FE000FE000FE0FFFE0FFFE0FFFEFFFE0F FFE0FFFE371F7E9E3C>I<07C07F0000FFC183C000FFC601E0000FC801F00007D000F000 07D000F80007E000F80007E000F80007C000F80007C000F80007C000F80007C000F80007 C000F80007C000F80007C000F80007C000F80007C000F80007C000F80007C000F80007C0 00F80007C000F80007C000F80007C000F80007C000F80007C000F80007C000F80007C000 F80007C000F8000FE001FC00FFFE1FFFC0FFFE1FFFC0221F7E9E27>I<001FE000007038 0001C00E00038007000F0003C00E0001C01C0000E03C0000F03C0000F078000078780000 7878000078F800007CF800007CF800007CF800007CF800007CF800007CF800007CF80000 7C780000787C0000F83C0000F03C0000F01E0001E01E0001E00F0003C00780078001C00E 0000703800001FE0001E1F7E9E23>I<07C1F0FFC618FFC83C0FD07E07D07E07E07E07E0 3C07E00007E00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C0 0007C00007C00007C00007C00007C00007C00007C00007C00007C0000FE000FFFF80FFFF 80171F7E9E1B>114 D<01FE080E01981800F8300078600038600038E00018E00018F000 18F80018FC00007FC0007FFE003FFF800FFFE007FFF000FFF80007FC0000FCC0003EC000 1EC0000EE0000EE0000EE0000EF0000CF8001CF80018E40030C300C080FF00171F7E9E1C >I<00600000600000600000600000600000E00000E00000E00001E00001E00003E00007 E0000FE0001FFFF0FFFFF003E00003E00003E00003E00003E00003E00003E00003E00003 E00003E00003E00003E00003E00003E00003E00003E00003E01803E01803E01803E01803 E01803E01803E01803E01801E01001F03000F020003840000F80152C7FAB1B>I<3FFFFF 803F001F003C003F0038003E0030007C003000FC007000F8006001F0006003F0006003E0 006007C000000FC000000F8000001F0000001F0000003E0000007C0000007C000000F801 8001F8018001F0018003E0018007E0018007C003800F8003001F8003001F0007003E0007 007E000F007C007F00FFFFFF00191F7E9E1F>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr12 35.8272 13 /Ft 13 119 df<000000000001E0000000000000000000000003F0000000000000000000 000003F0000000000000000000000003F0000000000000000000000007F8000000000000 000000000007F8000000000000000000000007F800000000000000000000000FFC000000 00000000000000000FFC00000000000000000000000FFC00000000000000000000001FFE 00000000000000000000001FFE00000000000000000000001FFE00000000000000000000 003FFF00000000000000000000003FFF00000000000000000000007FFF80000000000000 000000007FFF80000000000000000000007FFF8000000000000000000000FFFFC0000000 00000000000000F3FFC000000000000000000000F3FFC000000000000000000001F3FFE0 00000000000000000001E1FFE000000000000000000001E1FFE000000000000000000003 E1FFF000000000000000000003C0FFF000000000000000000003C0FFF000000000000000 000007C0FFF800000000000000000007807FF800000000000000000007807FF800000000 00000000000F007FFC0000000000000000000F003FFC0000000000000000000F003FFC00 00000000000000001E003FFE0000000000000000001E001FFE0000000000000000003E00 1FFF0000000000000000003C000FFF0000000000000000003C000FFF0000000000000000 007C000FFF800000000000000000780007FF800000000000000000780007FF8000000000 00000000F80007FFC00000000000000000F00003FFC00000000000000000F00003FFC000
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
0007FE000000FFC00007FE000000FFC00003FC000000FFC00000F0000000FFC000000000 0000FFC0000000000000FFC0000000000000FFC0000000000000FFC0000000000000FFC0 000000000000FFC00000000007FFFFC000000000FFFFFFC00000000FFFFFFFC00000007F FF00FFC0000001FFF000FFC0000007FF0000FFC000001FFC0000FFC000003FF00000FFC0 0000FFC00000FFC00001FF800000FFC00003FF000000FFC00007FE000000FFC0000FFC00 0000FFC0001FF8000000FFC0001FF8000000FFC0003FF0000000FFC0003FF0000000FFC0 077FE0000000FFC0077FE0000000FFC0077FE0000000FFC007FFC0000000FFC007FFC000 0000FFC007FFC0000000FFC007FFC0000001FFC007FFC0000001FFC007FFC0000001FFC0 07FFC0000003FFC0077FE0000003FFC0077FE00000077FC0077FE000000E7FC0073FF000 000E3FE00E1FF800001C3FE00E0FF80000781FF01C07FE0000F01FF83C03FF8003E00FFE F800FFE01F8007FFF0003FFFFF0003FFE0000FFFFC0001FFC00000FFE000007F00404379 C149>97 D<0000F800000000000007FFF8000000000000FFFFF8000000000000FFFFF800 0000000000FFFFF8000000000000FFFFF800000000000001FFF8000000000000007FF800 0000000000003FF8000000000000001FF8000000000000001FF8000000000000001FF800 0000000000001FF8000000000000001FF8000000000000001FF8000000000000001FF800 0000000000001FF8000000000000001FF8000000000000001FF8000000000000001FF800 0000000000001FF8000000000000001FF8000000000000001FF8000000000000001FF800 0000000000001FF8000000000000001FF8000000000000001FF8000000000000001FF800 0000000000001FF8000000000000001FF8000000000000001FF8000000000000001FF800 0000000000001FF8000000000000001FF8000000000000001FF8000000000000001FF800 0000000000001FF8000000000000001FF8000000000000001FF8000FFC000000001FF800 FFFFC00000001FF803FFFFF00000001FF80FE00FFC0000001FF81F0001FF0000001FF87C 00007F8000001FF8F000001FC000001FF9E000000FF000001FFB80000007F800001FFF00 000003FC00001FFF00000001FE00001FFE00000001FE00001FFC00000000FF00001FFC00 000000FF80001FF8000000007FC0001FF8000000007FC0001FF8000000003FE0001FF800 0000003FE0001FF8000000003FF0001FF8000000001FF0001FF8000000001FF8001FF800 0000001FF8001FF8000000001FF8001FF8000000001FFC001FF8000000000FFC001FF800 0000000FFC001FF8000000000FFC001FF8000000000FFE001FF8000000000FFE001FF800 0000000FFE001FF8000000000FFE001FF8000000000FFE001FF8000000000FFE001FF800 0000000FFE001FF8000000000FFE001FF8000000000FFE001FF8000000000FFE001FF800 0000000FFE001FF8000000000FFE001FF8000000000FFC001FF8000000000FFC001FF800 0000001FFC001FF8000000001FFC001FF8000000001FF8001FF8000000001FF8001FF800 0000001FF8001FF8000000003FF0001FF8000000003FF0001FF8000000003FE0001FF800 0000007FE0001FF8000000007FC0001FF8000000007F80001FFC00000000FF80001FFC00 000000FF00001FFE00000001FE00001FF700000003FC00001FE700000007F800001FC380 00000FF000001FC1C000001FE000001F80F000003FC000001F80780000FF0000001F003E 0003FE0000001E000FC01FF80000001E0007FFFFE00000001C0001FFFF80000000000000 1FF800000047687CE651>I<0000003FF80000000003FFFF000000000FFFFFE00000003F E00FF8000000FF0000FC000003FC00001F000007F800000780000FF0000003C0001FE000 0001C0003FC0000000E0007F80000007F000FF0000001FF001FF0000003FF003FE000000 3FF803FE0000007FF807FC0000007FF807FC0000007FF80FF80000007FF80FF80000003F F01FF80000003FF01FF00000001FE03FF000000007803FF000000000003FF00000000000 7FF000000000007FF000000000007FE000000000007FE00000000000FFE00000000000FF E00000000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE0 0000000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE000 000000007FE000000000007FF000000000007FF000000000007FF000000000003FF00000 0000003FF000000000003FF800000000001FF800000000001FF8000000000E0FFC000000 000E0FFC000000000E07FC000000001C03FE000000001C03FE000000003801FF00000000 3800FF8000000070007F8000000070003FC0000000E0001FE0000001C0000FF000000380 0007FC00000F000003FE00001E000000FF80007C0000007FF007F00000001FFFFFC00000 0003FFFF00000000003FF0000037437BC141>I<000000FFC00000000007FFFC00000000 3FFFFF00000000FF807FC0000001FE000FF0000007F80007F800000FF00001FE00001FE0 0000FF00003FC000007F00007F8000007F8000FF0000003FC001FF0000001FC001FE0000 001FE003FC0000001FF007FC0000000FF007FC0000000FF00FF80000000FF80FF8000000 07F81FF800000007F81FF000000007FC3FF000000007FC3FF000000003FC3FF000000003 FC7FF000000003FE7FE000000003FE7FE000000003FE7FE000000003FEFFE000000003FE FFE000000003FEFFFFFFFFFFFFFEFFFFFFFFFFFFFEFFFFFFFFFFFFFEFFE00000000000FF E00000000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE0 00000000007FE000000000007FE000000000007FE000000000007FF000000000007FF000
000000003FF000000000003FF000000000001FF000000000001FF8000000000E1FF80000 00000E0FF8000000000E0FF8000000001C07FC000000001C03FC000000001C03FE000000 003801FE000000007800FF0000000070007F80000000E0007FC0000001E0003FE0000003 C0000FF0000007800007F800001F000003FC00003E000000FF0001F80000003FE00FF000 00000FFFFFC000000003FFFE00000000003FF0000037437BC141>101 D<00000000000001FC00000001FF80001FFE0000000FFFF0007FFF0000007FFFFE01FE0F 800000FF00FF03E01FC00003FC003FCF801FC00007F0000FFE001FC0000FE00007FC001F C0001FC00003F8000F80003FC00003FC000700007F800001FE00000000FF800001FF0000 0000FF000000FF00000001FF000000FF80000001FF000000FF80000003FE0000007FC000 0003FE0000007FC0000003FE0000007FC0000007FE0000007FE0000007FE0000007FE000 0007FE0000007FE0000007FE0000007FE0000007FE0000007FE0000007FE0000007FE000 0007FE0000007FE0000007FE0000007FE0000007FE0000007FE0000003FE0000007FC000 0003FE0000007FC0000003FE0000007FC0000001FF000000FF80000001FF000000FF8000 0000FF000000FF00000000FF800001FF000000007F800001FE000000003FC00003FC0000 00001FC00003F8000000000FE00007F0000000001FF0000FE0000000003FFC003FC00000 000038FF00FF0000000000707FFFFE0000000000700FFFF00000000000F001FF80000000 0000E00000000000000001E00000000000000001E00000000000000001E0000000000000 0001E00000000000000001E00000000000000001F00000000000000001F0000000000000 0001F80000000000000001F80000000000000001FC0000000000000000FE000000000000 0000FF80000000000000007FFFFFFFC0000000007FFFFFFFFF000000003FFFFFFFFFE000 00001FFFFFFFFFF80000000FFFFFFFFFFE00000007FFFFFFFFFF0000000FFFFFFFFFFFC0 00003FFFFFFFFFFFE00000FF0000003FFFF00001FC00000001FFF00007F0000000003FF8 000FE0000000000FFC001FC00000000007FC001F800000000003FE003F800000000001FE 007F000000000000FE007F000000000000FE007F000000000000FF00FE0000000000007F 00FE0000000000007F00FE0000000000007F00FE0000000000007F00FE0000000000007F 00FE0000000000007F007E0000000000007E007F000000000000FE007F000000000000FE 003F800000000001FC001FC00000000003F8000FC00000000003F00007E00000000007E0 0003F8000000001FC00001FC000000003F800000FF00000000FF0000003FC0000003FC00 00001FF800001FF800000007FFC003FFE000000000FFFFFFFF00000000001FFFFFF80000 00000000FFFF0000000042617CC149>103 D<000F0000003FC000007FE00000FFF00000 FFF00001FFF80001FFF80001FFF80001FFF80000FFF00000FFF000007FE000003FC00000 0F0000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000F80003FFF8007FFFF8007FFFF8007FFFF8007F FFF80001FFF800007FF800003FF800001FF800001FF800001FF800001FF800001FF80000 1FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF80000 1FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF80000 1FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF80000 1FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF80000 1FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF80000 3FFC0000FFFF00FFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFE1F647BE329>105 D<0000F80007FFF800FFFFF800FFFFF800FFFFF800FFFFF80001FFF800007FF800003FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF800001FF8 00001FF800001FF800001FF800001FF800001FF800001FF800001FF800003FFC0000FFFF 00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF20677BE629>108 D<0001F0003FC00007FFF0 01FFF000FFFFF003FFF800FFFFF00FC0FC00FFFFF01F01FE00FFFFF03C03FF0001FFF078 07FF80007FF0F007FF80003FF0E007FF80001FF1C007FF80001FF3C007FF80001FF38003 FF00001FF78003FF00001FF70000FC00001FF700007800001FFE00000000001FFE000000 00001FFE00000000001FFC00000000001FFC00000000001FFC00000000001FFC00000000 001FFC00000000001FF800000000001FF800000000001FF800000000001FF80000000000 1FF800000000001FF800000000001FF800000000001FF800000000001FF800000000001F
F800000000001FF800000000001FF800000000001FF800000000001FF800000000001FF8 00000000001FF800000000001FF800000000001FF800000000001FF800000000001FF800 000000001FF800000000001FF800000000001FF800000000001FF800000000001FF80000 0000001FF800000000001FF800000000001FF800000000001FF800000000001FF8000000 00001FF800000000001FF800000000001FF800000000001FF800000000001FF800000000 001FFC00000000003FFC0000000000FFFF80000000FFFFFFFFE00000FFFFFFFFE00000FF FFFFFFE00000FFFFFFFFE0000031417CC039>114 D<0000FFF00060000FFFFE00E0003F FFFF81E000FF800FE3E001F80001F7E003E000007FE007C000003FE00F8000001FE01F00 00000FE03F00000007E03F00000003E07E00000003E07E00000003E07E00000001E0FE00 000001E0FE00000001E0FE00000000E0FF00000000E0FF00000000E0FF80000000E0FF80 000000E0FFC0000000E07FF0000000007FFC000000003FFF000000003FFFF80000001FFF FFC000000FFFFFFC00000FFFFFFF800007FFFFFFE00003FFFFFFF00000FFFFFFFC00007F FFFFFE00001FFFFFFF000007FFFFFF800000FFFFFFC0000007FFFFE00000003FFFE00000 0003FFF000000000FFF0000000003FF8600000001FF8E00000000FF8E000000007FCF000 000007FCF000000003FCF000000003FCF000000001FCF800000001FCF800000001FCF800 000001FCFC00000001FCFC00000001F8FE00000001F8FE00000001F8FF00000003F0FF00 000003F0FF80000007E0FFC0000007C0FFE000000FC0FEF000001F80FC7800003F00F83E 0000FE00F81FC00FF800F007FFFFE000E001FFFF8000C0003FF800002E437BC139>I<00 00F800000000F80007FFF8000007FFF800FFFFF80000FFFFF800FFFFF80000FFFFF800FF FFF80000FFFFF800FFFFF80000FFFFF80001FFF8000001FFF800007FF80000007FF80000 3FF80000003FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000001FF800001FF80000001FF80000 1FF80000001FF800001FF80000001FF800001FF80000003FF800001FF80000003FF80000 1FF80000003FF800001FF80000003FF800001FF80000007FF800001FF80000007FF80000 0FF80000007FF800000FF8000000FFF800000FF8000000FFF800000FFC000001DFF80000 07FC0000039FF8000007FC0000039FF8000003FC0000071FFC000001FE00000E1FFE0000 00FF00003C1FFF8000007FC000781FFFFF00003FF803F01FFFFF00000FFFFFC01FFFFF00 0003FFFF001FFFFF0000003FF8001FFC0048427BC051>117 D<FFFFFFFC0000FFFFFCFF FFFFFC0000FFFFFCFFFFFFFC0000FFFFFCFFFFFFFC0000FFFFFC01FFFF0000003FFFC000 7FFC0000000FFF00003FFC00000007FC00003FF800000003F800001FF800000003F00000 1FFC00000001E000000FFC00000001E000000FFC00000001C000000FFE00000003C00000 07FE0000000380000007FF0000000380000003FF0000000700000003FF00000007000000 03FF8000000F00000001FF8000000E00000001FFC000000E00000000FFC000001C000000 00FFC000001C00000000FFE000003C000000007FE0000038000000007FF0000038000000 003FF0000070000000003FF0000070000000003FF80000F0000000001FF80000E0000000 001FFC0000E0000000000FFC0001C0000000000FFC0001C0000000000FFE0003C0000000 0007FE0003800000000007FF0003800000000003FF0007000000000003FF000700000000 0003FF800F000000000001FF800E000000000001FFC00E000000000000FFC01C00000000 0000FFC01C000000000000FFE03C0000000000007FE0380000000000007FF03800000000 00003FF0700000000000003FF0700000000000003FF8F00000000000001FF8E000000000 00001FFCE00000000000000FFDC00000000000000FFDC00000000000000FFFC000000000 000007FF8000000000000007FF8000000000000003FF0000000000000003FF0000000000 000003FF0000000000000001FE0000000000000001FE0000000000000000FC0000000000 000000FC0000000000000000FC0000000000000000780000000000000000300000000046 417DBF4D>I E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin
%%EndSetup %%Page: 1 1 1 0 bop 431 735 a Ft(Algebraic)46 b(Curv)l(es)537 829 y Fs(Professor:)23 b(H.)8 b(W.)23 b(Lenstra,)g(Jr.)509 907 y(Math)h(274,)g(F)-6 b(all)22 b(Semester,)h(1995)595 986 y(Head)g(Scrib)r(e:)h(Ezra)g(Miller)283 1764 y Fr(The)d(follo)o (wing)f(notes)h(are)f(from)h(the)g(class)f Fq(A)o(lgebr)m(aic)i(Curves) f Fr(giv)o(en)283 1834 y(b)o(y)f(Hendrik)f(Lenstra)g(at)h(U.C.)f(Berk)o (eley)h(in)f(the)h(fall)g(of)f(1995.)g(They)283 1904 y(are)13 b(a)g(concatenation)g(of)g(informal)f(class)g(notes)h(tak)o (en)h(b)o(y)e(the)i(studen)o(ts)283 1974 y(in)k(the)h(class.)e(They)h (ha)o(v)o(e)g(not)g(b)q(een)g(read)g(b)o(y)f(Professor)g(Lenstra)h (\(or)283 2043 y(Ezra)j(Miller\),)e(and)h(Prof.)g(Lenstra)g(tak)o(es)h (no)f(resp)q(onsibilit)o(y)f(for)h(the)283 2113 y(con)o(ten)o(ts.)963 2790 y(1)p eop %%Page: 2 2 2 1 bop 0 50 a Fp(Math)19 b(274:)f(Algebraic)j(curv)n(es.)0 120 y Fr(H.)8 b(W.)13 b(Lenstra,)f(Jr.)f(\(879)i(Ev)m(ans,)f(o\016ce)g (hours)f(MF)h(4:15{5:15,)f(tel.)i(643{7857,)e(e-mail)g(h)o(wl@math\).)0 189 y(F)l(all)16 b(1995,)g(MWF)g(3{4,)g(65)g(Ev)m(ans.)0 299 y(This)e(is)h(an)g(in)o(tro)q(ductory)f(course)g(on)g(algebraic)g (curv)o(es)h(o)o(v)o(er)f(arbitrary)g(\014elds.)f(It)j(is)f(exp)q (ected)h(that)0 369 y(it)f(will)f(b)q(e)g(taugh)o(t)g(regularly)g(in)g (the)g(future,)g(as)g(Math)g(255.)g(The)h(course)e(o)q(ccupies)h(an)g (in)o(termediate)0 438 y(p)q(osition)20 b(b)q(et)o(w)o(een)h(Math)f (254)h(\(Num)o(b)q(er)f(theory\))h(and)g(Math)f(256)h(\(Algebraic)f (geometry\).)h(It)h(is)0 508 y(recommended)16 b(for)i(studen)o(ts)f (who)h(lik)o(e)g(to)g(acquire)g(a)g(go)q(o)q(d)g(understanding)d(of)k (a)f(sp)q(ecial)g(class)f(of)0 578 y(v)m(arieties)e(b)q(efore)g(em)o (barking)e(up)q(on)h(the)h(more)f(general)g(and)h(abstract)f(approac)o (h)f(of)i(sc)o(heme)f(theory)0 648 y(that)24 b(is)e(o\013ered)h(in)f (Math)h(256.)g(Another)g(ma)s(jor)f(di\013erence)g(with)h(Math)g(256)g (is)g(that)g(w)o(e)g(shall)0 717 y(w)o(ork)16 b(o)o(v)o(er)f(base)h (\014elds)f(that)i(are)e(not)i(necessarily)e(algebraically)g(closed,)g (including)g(n)o(um)o(b)q(er)f(\014elds)0 787 y(and)j(\014nite)g (\014elds.)g(This)g(will)g(mak)o(e)g(the)h(course)f(attractiv)o(e)h (for)g(studen)o(ts)e(with)i(an)f(in)o(terest)h(in)f(the)0 857 y(arithmetic)d(asp)q(ects)h(of)g(algebraic)f(geometry)l(.)h (Another)f(link)h(with)g(n)o(um)o(b)q(er)e(theory)i(is)f(found)g(in)h (the)0 927 y(structural)g(analogy)h(b)q(et)o(w)o(een)h(function)f (\014elds)f(of)i(curv)o(es)f(and)g(n)o(um)o(b)q(er)e(\014elds.)100 996 y(I)19 b(exp)q(ect)h(to)f(co)o(v)o(er)f(the)h(follo)o(wing)f (topics:)g(algebraic)g(curv)o(es,)g(singularities,)f(function)h (\014elds,)0 1066 y(v)m(aluations,)f(line)g(bundles,)f(the)i(theorem)g (of)g(Riemann-Ro)q(c)o(h,)d(Hurwitz's)i(form)o(ula,)f(curv)o(es)h(of)h (lo)o(w)0 1136 y(gen)o(us,)c(elliptic)h(curv)o(es)f(\(merely)i(scratc)o (hing)e(the)h(surface)g(here!\),)g(h)o(yp)q(erelliptic)f(curv)o(es,)g (pro)s(jectiv)o(e)0 1205 y(em)o(b)q(eddings,)i(the)j(canonical)e(curv)o (e,)g(zeta)i(functions)f(of)g(curv)o(es)g(o)o(v)o(er)f(\014nite)h (\014elds;)f(in)h(addition,)f(I)0 1275 y(hop)q(e)f(to)g(address)e(a)i (selection)g(of)g(the)g(follo)o(wing:)f(the)h(Riemann)f(h)o(yp)q (othesis)f(for)i(curv)o(es)f(o)o(v)o(er)g(\014nite)0 1345 y(\014elds,)g(applications)g(to)i(co)q(ding)f(theory)l(,)h (construction)e(of)i(Jacobian)e(v)m(arieties.)100 1415 y(Eac)o(h)24 b(studen)o(t)h(who)f(wishes)h(to)g(obtain)g(credit)g(for)g (the)g(course)f(m)o(ust)g(participate)h(in)g(the)0 1484 y(\\scrib)q(e")e(system,)g(whereb)o(y)f(eac)o(h)i(lecture)f(is)g (transcrib)q(ed)f(in)h(plain)g(T)1430 1495 y(E)1457 1484 y(X)h(and)f(made)g(a)o(v)m(ailable)0 1554 y(to)16 b(the)g(other)f
(studen)o(ts)f(\(and,)h(p)q(ossibly)l(,)f(to)i(future)f(instructors)f (of)i(Math)f(255\).)h(In)f(addition,)f(there)0 1624 y(will)i(b)q(e)h (regular)e(homew)o(ork)g(assignmen)o(ts.)f(There)i(will)g(not)h(b)q(e)g (a)f(\014nal)g(examination.)100 1694 y(Prerequisites)h(for)h(the)h (course)f(are)g(Math)g(250A)h(and)f(either)g(Math)g(250B)h(or)f(the)h (consen)o(t)f(of)0 1763 y(the)i(instructor.)d(F)l(amiliarit)o(y)h(with) h(the)h(basic)e(prop)q(erties)g(of)i(groups,)d(rings,)h(and)h(\014elds) f(is)h(essen-)0 1833 y(tial.)c(Studen)o(ts)f(not)g(acquain)o(ted)h (with)f(sligh)o(tly)g(more)g(adv)m(anced)h(sub)s(jects)f(suc)o(h)g(as)h (mo)q(dules,)e(exact)0 1903 y(sequences,)i(lo)q(calization,)f(tensor)h (pro)q(ducts,)f(Galois)g(theory)l(,)h(transcendence)f(degree,)h(and)g (the)g(lan-)0 1973 y(guage)e(\()p Fq(only)19 b Fr(the)14 b(language!\))f(of)h(categories,)f(are)g(exp)q(ected)i(to)f(b)q(e)g (willing)f(and)g(able)g(to)h(learn)f(ab)q(out)0 2042 y(these)i(on)g(the)g(side.)g(More)f(adv)m(anced)h(topics)g(from)f (algebra)h(that)g(are)g(needed)g(will)f(b)q(e)i(dev)o(elop)q(ed)e(in)0 2112 y(class,)j(including)g(v)m(aluation)h(theory)l(.)g(Prior)f(exp)q (osure)h(to)h(algebraic)e(n)o(um)o(b)q(er)f(theory)i(or)g(algebraic)0 2182 y(geometry)e(will)g(not)h(h)o(urt,)e(but)i(will)f(not)g(b)q(e)h (assumed.)100 2252 y(Some)10 b(of)h(the)h(literature)e(on)h(the)g(sub)s (ject,)g(suc)o(h)f(as)h(Hartshorne)f(and)g(F)l(ulton)g(b)q(elo)o(w,)h (emphasizes)0 2321 y(the)18 b(geometric)g(asp)q(ects)f(and)h(places)f (the)h(sub)s(ject)g(in)f(the)h(general)f(con)o(text)i(of)f(solving)f (systems)g(of)0 2391 y(p)q(olynomial)12 b(equations,)g(often)h(o)o(v)o (er)f(algebraically)g(closed)g(\014elds.)g(Other)h(b)q(o)q(oks,)g(suc)o (h)e(as)i(Artin)g(and)0 2461 y(Stic)o(h)o(tenoth,)j(deal)h(primarily)e (with)i(the)h(algebraic)e(asp)q(ects,)h(notably)g(the)g(arithmetic)g (of)g(function)0 2531 y(\014elds.)h(In)h(the)g(presen)o(t)f(course)g(I) h(will)f(attempt)i(to)f(do)g(justice)g(to)g(b)q(oth)g(p)q(oin)o(ts)f (of)h(view,)h(m)o(uc)o(h)d(of)0 2600 y(the)f(motiv)m(ation)g(b)q(eing)g (geometric)f(and)h(most)f(pro)q(of)h(tec)o(hniques)f(algebraic.)g (There)g(is)h(no)f(required)0 2670 y(textb)q(o)q(ok.)963 2790 y(2)p eop %%Page: 3 3 3 2 bop 100 50 a Fr(Literature:)100 120 y(R.)14 b(Hartshorne,)g Fo(Algebraic)g(geometry)p Fr(,)h(Springer-V)l(erlag,)d(New)k(Y)l(ork,)f (1977,)f(Chapters)g(I)h(\(es-)0 190 y(p)q(ecially)i(section)f(6\))h (and)f(IV.)100 260 y(W.)f(F)l(ulton,)e Fo(Algebraic)h(curv)o(es,)g(an)h (in)o(tro)q(duction)e(to)j(algebraic)d(geometry)p Fr(,)i(Benjamin,)f (Read-)0 330 y(ing,)i(Mass.,)f(1969.)100 400 y(E.)g(Artin,)g Fo(Algebraic)g(n)o(um)o(b)q(ers)f(and)h(algebraic)f(functions)p Fr(,)h(Gordon)f(and)h(Breac)o(h,)g(New)h(Y)l(ork,)0 470 y(1967.)100 540 y(H.)g(Stic)o(h)o(tenoth,)g Fo(Algebraic)g(function)g (\014elds)g(and)f(co)q(des)p Fr(,)i(Springer-V)l(erlag,)c(Berlin,)j (1993.)100 610 y(More)f(literature)h(ma)o(y)g(b)q(e)h(giv)o(en)f(in)g (class.)0 740 y Fp(Lecture)k(1)c Fr(\(August)h(28,)f(1995\).)0 810 y(Scrib)q(e:)g(Da)o(vid)g(Jones)0 942 y Fp(Ho)n(w)j(do)r(es)f(this) i(course)f(\(math)f(255?\))g(compare)f(with)k(math)d(254)g(and)g(256?) 100 1015 y Fr(Math)e(256)g(deals)g(primarily)e(with)j(algebraically)e (closed)h(\014elds,)g Fn(k)f Fr(=)p 1453 974 28 2 v 14 w Fn(k)q Fr(.)100 1085 y(Math)g(254)g(deals)g(primarily)f(with)h (\014nite)g(\014elds)g(and)g(algebraic)f(n)o(um)o(b)q(er)g(\014elds)g (\([)p Fn(k)i Fr(:)e Fp(Q)p Fr(])f Fn(<)h Fm(1)p Fr(\).)100 1155 y(This)h(class)h(will)g(deal)g(with)h Fp(F)-5 b(unction)20 b(\014elds)g(of)e(algebraic)i(curv)n(es)p Fr(,)d Fn(K)q(=k)r Fr(.)100 1265 y(In)h(this)g(course)g(w)o(e)h(will)f(see)h(the)g (corresp)q(ondence)e(b)q(et)o(w)o(een)h(algebraic)g(curv)o(es)f(and)h (function)0 1335 y(\014elds.)11 b(Algebraic)g(curv)o(es)g(will)g(b)q(e) i(v)m(ague)f(and)f(in)o(tuitiv)o(e)g(ob)s(jects.)h(F)l(unction)e
(\014elds)h(will)h(b)q(e)g(rigorously)0 1404 y(dev)o(elop)q(ed.)100 1515 y Fp(Basic)19 b(ob)s(ject)f(of)g(study)p Fr(.)d(Fix)h(a)g(\014eld) f Fn(k)r Fr(.)h(A)g Fp(function)j(\014eld)d Fr(\(of)h(a)f(curv)o(e)f(o) o(v)o(er)g Fn(k)r Fr(\))h(is)g(a)g(\014eld)0 1584 y(extension)g Fn(K)21 b Fr(with)16 b Fn(k)g Fm(\032)d Fn(K)21 b Fr(suc)o(h)15 b(that:)170 1654 y(1.)i Fn(K)j Fr(is)c(\014nitely)h(generated)f(as)g(a) h(\014eld)e(extension)i(of)g Fn(k)r Fr(;)170 1725 y(2.)g(tr.deg\()p Fn(K)q(=k)r Fr(\))c(=)h(1;)i(\(i.e.)h Fm(9)p Fn(t)c Fm(2)h Fn(K)21 b Fr(s.t.)16 b Fn(t)h Fr(is)f(not)g(alg./)p Fn(k)i Fr(and)e Fn(K)k Fr(is)c(alg.)g(/)p Fn(k)r Fr(\()p Fn(t)p Fr(\).\))170 1797 y(3.)p 226 1757 V 17 w Fn(k)c Fm(\\)g Fn(K)17 b Fr(=)d Fn(k)k Fr(\(inside)p 626 1757 46 2 v 16 w Fn(K)t Fr(\).)100 1907 y(Let)f Fn(k)h Fr(b)q(e)f(an)o(y)f (\014eld.)100 2030 y(\\Curv)o(e)f(o)o(v)o(er)h Fn(k)r Fr(")447 2001 y Fl(def)455 2030 y Fr(=)24 b(a)17 b(one-dimensional)d(v) m(ariet)o(y)i(o)o(v)o(er)g Fn(k)r Fr(.)100 2100 y(But)h(what)f(are)g(v) m(arieties)h(and)f(what)g(do)q(es)h(dimension)d(mean?)100 2210 y Fp(F)-5 b(or)18 b(no)n(w)i(w)n(e)g(will)i(assume)17 b Fn(k)e Fr(=)p 807 2169 28 2 v 14 w Fn(k)q Fr(.)100 2280 y(An)d Fp(algebraic)j(set)d Fr(is)f(a)h(subset)g Fn(V)23 b Fr(of)12 b Fn(k)860 2262 y Fk(n)899 2280 y Fr(\(for)g(some)f Fn(n)j Fm(\025)f Fr(0\))f(for)g(whic)o(h)f(there)h (exist)g Fn(f)1732 2287 y Fl(1)1755 2280 y Fn(;)c(:::;)g(f)1865 2287 y Fk(m)1917 2280 y Fm(2)0 2350 y Fn(k)r Fr([)p Fn(X)83 2357 y Fl(1)105 2350 y Fn(;)g(:::;)g(X)232 2357 y Fk(n)259 2350 y Fr(])17 b(suc)o(h)e(that)i Fn(V)25 b Fr(=)13 b Fm(f)p Fn(x)i Fr(=)e(\()p Fn(x)783 2357 y Fk(i)801 2350 y Fr(\))820 2332 y Fk(n)820 2363 y(i)p Fl(=1)901 2350 y Fm(2)h Fn(k)976 2332 y Fk(n)1016 2350 y Fm(j)g Fn(f)1068 2357 y Fl(1)1091 2350 y Fr(\()p Fn(x)p Fr(\))h(=)e Fn(:::)h Fr(=)f Fn(f)1356 2357 y Fk(m)1395 2350 y Fr(\()p Fn(x)p Fr(\))i(=)e(0)p Fm(g)p Fr(.)100 2420 y(Note:)k Fn(V)28 b Fr(is)16 b(empt)o(y)g(i\013)g(some)g(linear)g(com)o(bination)e(of)j (the)g Fn(f)1266 2427 y Fk(i)1283 2420 y Fr('s)f(giv)o(es)g(1.)100 2490 y(Note:)h Fn(x)262 2472 y Fl(2)262 2503 y(1)296 2490 y Fr(+)11 b Fn(x)374 2472 y Fl(2)374 2503 y(2)407 2490 y Fr(+)g(1)j(=)g(0)i(has)g(solutions)f(o)o(v)o(er)h Fp(C)h Fr(but)f(none)g(o)o(v)o(er)g Fp(R)g Fr(so)g Fn(k)g Fr(=)p 1583 2450 V 13 w Fn(k)i Fr(is)f(imp)q(ortan)o(t.)100 2600 y(A)j Fp(morphism)d Fn(V)31 b Fm(!)19 b Fn(V)596 2582 y Fj(0)630 2600 y Fr(is)h(a)g(map)f(\\giv)o(en)g(b)o(y)h Fn(n)1104 2582 y Fj(0)1138 2600 y Fr(p)q(olynomials,)e Fn(g)1456 2607 y Fl(1)1478 2600 y Fn(;)8 b(:::;)g(g)1588 2607 y Fk(n)1613 2598 y Fi(0)1648 2600 y Fr(in)20 b Fn(n)g Fr(v)m(ariables)0 2670 y(o)o(v)o(er)c Fn(k)r Fr(.")963 2790 y(3)p eop %%Page: 4 4 4 3 bop 100 50 a Fr(A)o(t)15 b(this)g(p)q(oin)o(t,)g(w)o(e)g(ha)o(v)o (e)g(a)g(category)l(.)h(Sp)q(eci\014cally)l(,)e(w)o(e)i(ha)o(v)o(e)e (the)i(notion)f(of)g(an)g(isomorphism:)0 120 y(a)g(morphism)e(suc)o(h)h (that)i(there)f(is)g(another)g(morphism)e(in)i(the)g(opp)q(osite)g (direction)g(so)g(that)h(comp)q(o-)0 189 y(sition)g(of)h(the)f(t)o(w)o (o)g(in)h(either)f(direction)g(is)g(the)g(iden)o(tit)o(y)l(.)100 302 y Fp(Ex:)g Fr(Let)h Fn(V)25 b Fr(:)14 b Fn(y)397 284 y Fl(2)433 302 y Fr(=)g Fn(x)514 284 y Fl(3)548 302 y Fm(\000)d Fn(x)p Fr(,)16 b Fn(V)697 284 y Fj(0)724 302 y Fr(=)e Fn(k)805 284 y Fl(1)827 302 y Fr(.)i(De\014ne)h(a)f(map)g Fn(V)25 b Fm(!)13 b Fn(V)1322 284 y Fj(0)1352 302 y Fr(b)o(y)j(\()p Fn(x;)8 b(y)r Fr(\))16 b Fm(7!)d Fn(x)p Fr(.)100 375
y(This)i(morphism)f(is)i(certainly)h(not)f(an)g(isomorphism)e(b)q (ecause)i(it)h(is)f(not)g(ev)o(en)h(bijectiv)o(e.)100 448 y Fp(Ex:)d Fr(Let)i Fn(V)25 b Fr(:)14 b Fn(y)i Fr(=)d Fn(x)488 430 y Fl(2)511 448 y Fr(,)i Fn(V)580 430 y Fj(0)608 448 y Fr(=)f Fn(k)689 430 y Fl(1)711 448 y Fr(.)g(De\014ne)h(a)g(map)f Fn(V)25 b Fm(!)14 b Fn(V)1199 430 y Fj(0)1228 448 y Fr(b)o(y)h(\()p Fn(x;)8 b(y)r Fr(\))15 b Fm(7!)f Fn(x)p Fr(,)h(and)f(de\014ne)h (another)0 518 y(map)h(b)o(y)g Fn(V)220 499 y Fj(0)248 518 y Fm(!)d Fn(V)28 b Fr(b)o(y)16 b Fn(x)f Fm(7!)e Fr(\()p Fn(x;)8 b(x)640 499 y Fl(2)664 518 y Fr(\).)100 590 y(These)15 b(t)o(w)o(o)h(morphisms)d(are)j(in)o(v)o(erses)f(so)h(eac)o(h)f(is)h (an)g(isomorphism)d(and)j(w)o(e)g(should)e(view)j(the)0 660 y(t)o(w)o(o)f(v)m(arieties)h(as)f(b)q(eing)g(the)h(same)f(ob)s (ject.)100 773 y(W)l(e)f(will)g(ev)o(en)o(tually)g(attac)o(h)g(in)o(v)m (arian)o(ts)e(to)j(these)f(v)m(arieties:)g(one)h(example)e(is)h(the)h (gen)o(us.)e(Tw)o(o)0 843 y(isomorphic)g(v)m(arieties)j(will)f(then)g (ha)o(v)o(e)g(the)h(same)f(genera.)100 955 y(Fix)j(some)h(v)m(ariet)o (y)g Fn(V)11 b Fr(.)20 b(The)g Fp(co)r(ordinate)j(ring)d Fn(A)p Fr(\()p Fn(V)12 b Fr(\))20 b(of)g Fn(V)31 b Fr(is)20 b(the)g(ring)f(of)h(all)g(morphisms)0 1025 y Fn(f)g Fr(:)13 b Fn(V)25 b Fm(!)14 b Fn(k)217 1007 y Fl(1)239 1025 y Fr(.)100 1098 y(Ev)o(ery)23 b Fn(f)32 b Fm(2)26 b Fn(A)p Fr(\()p Fn(V)11 b Fr(\))25 b(is)e(a)g(p)q(olynomial)g(in)g(the)h Fn(n)g Fr(co)q(ordinates)f Fn(x)1387 1105 y Fl(1)1409 1098 y Fn(;)8 b(:::x)1501 1105 y Fk(n)1529 1098 y Fr(.)23 b(\(Hence)i(the)f(name)0 1168 y(\\co)q(ordinate)16 b(ring".\))100 1241 y(So)i(w)o(e)g(get)g(a)g(surjectiv)o(e)g(map)g Fn(k)r Fr([)p Fn(X)794 1248 y Fl(1)816 1241 y Fn(;)8 b(:::;)g(X)943 1248 y Fk(n)970 1241 y Fr(])16 b Fm(!)h Fn(A)p Fr(\()p Fn(V)12 b Fr(\).)18 b(In)g(fact,)h(since)f(the)g Fn(f)1628 1248 y Fk(i)1645 1241 y Fr('s)g(are)g(0)g(on)g Fn(V)11 b Fr(,)0 1310 y(w)o(e)16 b(get)h(a)g(surjection)f Fn(k)r Fr([)p Fn(X)510 1317 y Fl(1)532 1310 y Fn(;)8 b(:::;)g(X)659 1317 y Fk(n)686 1310 y Fr(])p Fn(=)p Fr(\()p Fn(f)768 1317 y Fl(1)791 1310 y Fn(;)g(:::;)g(f)901 1317 y Fk(m)939 1310 y Fr(\))14 b Fm(!)f Fn(A)p Fr(\()p Fn(V)f Fr(\).)100 1383 y(The)k(question)g(w)o(e)h(w)o(an)o(t)e(to)i(answ)o(er)e(is...)100 1456 y Fp(Is)j(this)i(map)d(an)i(isomorphism?)100 1569 y Fr(Unfortunately)l(,)d(the)h(answ)o(er)e(is)h(no.)100 1643 y(Let)i Fn(R)h Fr(b)q(e)f(a)g(comm)o(utativ)o(e)f(ring,)f(and)i (let)g Fn(I)i Fm(\032)c Fn(R)i Fr(b)q(e)g(an)g(ideal.)f(Then)1506 1601 y Fm(p)p 1547 1601 26 2 v 1547 1643 a Fn(I)j Fr(=)c Fm(f)p Fn(r)i Fm(2)f Fn(R)f Fm(j)g(9)p Fn(n)g Fm(2)0 1713 y Fp(Z)35 1720 y Fj(\025)p Fl(0)105 1713 y Fr(s)p Fn(:)p Fr(t)p Fn(:)g(r)211 1694 y Fk(n)253 1713 y Fm(2)e Fn(I)t Fm(g)p Fr(.)100 1791 y(This)h(de\014nition)h(giv)o(es)g(us)g(a)g (map)g Fn(k)r Fr([)p Fn(X)856 1798 y Fl(1)878 1791 y Fn(;)8 b(:::;)g(X)1005 1798 y Fk(n)1032 1791 y Fr(])p Fn(=)1071 1749 y Fh(p)p 1121 1749 234 2 v 42 x Fr(\()p Fn(f)1164 1798 y Fl(1)1187 1791 y Fn(;)g(:::;)g(f)1297 1798 y Fk(m)1335 1791 y Fr(\))1363 1763 y Fj(\030)1354 1791 y Fm(!)p Fn(A)p Fr(\()p Fn(V)k Fr(\).)100 1904 y(Giv)o(en)24 b(a)g(morphism)e Fn(V)38 b Fm(!)26 b Fn(V)725 1886 y Fj(0)739 1904 y Fr(,)f(w)o(e)f(get)g(a)h(homomorphism)20 b(\(ring)k(hom.,)f(iden)o(tit)o(y)h(on)g Fn(k)r Fr(\))0 1974 y Fn(A)p Fr(\()p Fn(V)97 1956 y Fj(0)111 1974 y Fr(\))14 b Fm(!)g Fn(A)p Fr(\()p Fn(V)e Fr(\))17 b(giv)o(en)f(b)o(y)g Fn(X)581 1981 y Fk(i)611 1974 y Fm(7!)e Fn(g)699 1981 y Fk(i)715 1974 y Fr(\()p Fn(X)775 1981 y Fl(1)798 1974
y Fn(;)8 b(:::;)g(X)925 1981 y Fk(n)953 1974 y Fr(\).)100 2086 y(W)l(e)16 b(also)g(w)o(an)o(t)g(our)g(algebraic)f(sets)i(to)g(b)q (e)f Fo(irreducible)p Fr(.)f(What)i(do)q(es)f(this)g(mean?)100 2159 y(A)f(non-empt)o(y)e(algebraic)g(set)i Fn(V)25 b Fm(\032)13 b Fn(k)817 2141 y Fk(n)859 2159 y Fr(is)h Fp(reducible)i Fr(if)f(there)f(are)g(algebraic)g(sets)g Fn(V)1733 2166 y Fl(1)1755 2159 y Fn(;)8 b(V)1806 2166 y Fl(2)1843 2159 y Fm(\032)13 b Fn(k)1923 2141 y Fk(n)0 2229 y Fr(di\013eren)o(t)j(from)f Fn(V)28 b Fr(suc)o(h)15 b(that)i Fn(V)25 b Fr(=)14 b Fn(V)727 2236 y Fl(1)760 2229 y Fm([)e Fn(V)834 2236 y Fl(2)856 2229 y Fr(.)17 b(Otherwise,)e Fn(V)28 b Fr(is)16 b Fp(irreducible)p Fr(.)100 2302 y(There)h(is)g(a)h(theorem)f(\(dep)q(ending)f(on)i(the)g (p)q(olynomial)e(ring)h(b)q(eing)g(No)q(etherian\))h(that)g(ev)o(ery)0 2372 y(algebraic)d(set)i(can)g(b)q(e)f(written)h(as)f(a)h(\014nite)f (union)f(of)i(irreducible)d(algebraic)i(sets.)100 2445 y(An)g(irreducible)f(algebraic)g(set)i(is)f(called)g(an)g Fp(a\016ne)j(v)m(ariet)n(y)f Fr(\(o)o(v)o(er)e Fn(k)r Fr(\).)100 2557 y Fo(Exercise:)g Fn(V)27 b Fr(is)16 b(irreducible)29 b Fm(\()-8 b(\))27 b Fn(A)p Fr(\()p Fn(V)12 b Fr(\))17 b(is)f(a)g(domain.)100 2670 y(Next)h(time)g(w)o(e)f(will)g(consider:) 963 2790 y(4)p eop %%Page: 5 5 5 4 bop 460 50 a Fr(1.)16 b(Non-algebraically)f(closed)h(\014elds.)460 120 y(2.)g(Dimensions)f(of)h(v)m(arieties)460 190 y(3.)g(Non-a\016ne)f (v)m(arieties.)0 320 y Fp(Lecture)20 b(2)c Fr(\(August)h(30,)f(1995\).) 0 390 y(Scrib)q(e:)g(Ezra)g(Miller)0 520 y(As)h(usual)e(w)o(e)h(let)h Fn(k)h Fr(b)q(e)f(a)g(\014eld)f(and)f Fn(K)21 b Fr(b)q(e)c(a)f (function)g(\014eld)g(o)o(v)o(er)g Fn(k)r Fr(.)100 630 y(Consider)11 b(the)j(case)f(when)g Fn(k)i Fr(=)p 710 590 28 2 v 14 w Fn(k)q Fr(.)f(Then)e(the)i(set)f(of)h(\(isomorphism)c (classes)j(of)t(\))g(a\016ne)g(v)m(arieties)0 700 y(corresp)q(onds)21 b(bijectiv)o(ely)j(with)f(the)g(set)h(of)f(\(isomorphism)d(classes)i (of)t(\))i(domains)d(of)j(\014nite)f(t)o(yp)q(e)0 770 y(o)o(v)o(er)18 b Fn(k)r Fr(.)h(In)f(fact,)i(the)f(function)g(whic)o(h) f(to)h(eac)o(h)g(a\016ne)f(v)m(ariet)o(y)i(asso)q(ciates)e(its)h (function)g(\014eld)f(is)h(a)0 839 y(con)o(tra)o(v)m(arian)o(t)d (functor,)i(so)g(a)g(morphism)e Fn(V)28 b Fm(!)17 b Fn(V)1000 821 y Fj(0)1032 839 y Fr(corresp)q(onds)f(to)j(a)f(homomorphism)d Fn(A)p Fr(\()p Fn(V)d Fr(\))17 b Fm( )0 909 y Fn(A)p Fr(\()p Fn(V)97 891 y Fj(0)111 909 y Fr(\))24 b(\014xing)e Fn(k)r Fr(.)h(\(\\Con)o(tra)o(v)m(arian)o(t")f(means)g(that)i(the)f (arro)o(ws)f(ha)o(v)o(e)g(b)q(een)i(rev)o(ersed.\))e(W)l(e)i(no)o(w)0 979 y(consider)15 b(what)i(happ)q(ens)e(in)h(the)h(three)f(cases)g(men) o(tioned)g(at)h(the)f(end)g(of)h(Lecture)g(1.)0 1089 y Fp(1.)j(General)h Fn(k)r Fr(:)c(W)l(e)h(relax)f(the)h(conditions)f (on)g Fn(k)r Fr(,)g(and)g(no)g(longer)g(require)g(it)g(to)h(b)q(e)g (algebraically)0 1158 y(closed.)c(It)i(is)f(p)q(ossible)f(no)o(w)g(to)i (get)g(di\013eren)o(t)e(equations)h(o)o(v)o(er)f Fn(k)j Fr(whic)o(h,)d(from)h(the)g(p)q(oin)o(t)g(of)g(view)h(of)0 1228 y(solutions)h(in)h Fn(k)r Fr(,)f(lo)q(ok)i(exactly)g(the)f(same.)g (Ho)o(w)o(ev)o(er,)f(w)o(e)h(don't)g(w)o(an)o(t)f(to)h(call)g(t)o(w)o (o)g(curv)o(es)f(suc)o(h)g(as)0 1298 y Fn(x)28 1280 y Fl(2)58 1298 y Fr(+)7 b Fn(y)130 1280 y Fl(2)158 1298 y Fr(+)g(1)13 b(=)h(0)g(and)g Fn(x)457 1280 y Fl(4)486 1298 y Fr(+)7 b Fn(y)558 1280 y Fl(4)587 1298 y Fr(+)g(1)13 b(=)h(0)g(isomorphic)e(simply)h(b)q(ecause)i Fn(k)g Fr(is)f(to)q(o)h (small)e(to)i(get)g(enough)0 1367 y(solutions)f(\(the)i(example)f(to)h (b)q(e)g(thinking)e(ab)q(out)i(is)f Fn(k)g Fr(=)f Fp(R)p
Fr(\).)h(There)g(are)g(a)h(n)o(um)o(b)q(er)d(of)j(suggestions)0 1437 y(w)o(e)g(could)g(en)o(tertain)g(to)h(try)g(to)g(squeeze)f(out)h (an)f(appropriate)f(de\014nition)g(of)i(a)f(v)m(ariet)o(y)l(.)100 1507 y Fq(Suggestion)i(1.)f Fr(Use)g(as)f(a)g(de\014nition)g(the)h(one) f(in)o(v)o(olving)f(domains)g(of)h(\014nite)h(t)o(yp)q(e)g(o)o(v)o(er)f Fn(k)r Fr(.)100 1577 y Fq(Suggestion)j(2.)f Fr(Agree)g(that)g(the)g(co) q(e\016cien)o(ts)f(of)h(y)o(our)e(p)q(olynomial)h(lie)g(in)g Fn(k)r Fr(,)g(while)h(y)o(our)e(zero)0 1647 y(sets)g(lie)h(in)p 222 1606 V 16 w Fn(k)q Fr(.)100 1717 y Fq(Suggestion)f(3.)f Fr(View)g(the)g(zeros)f(as)g(lying)g(in)g Fq(any)i Fr(extension)e (\014eld)g Fn(L=k)r Fr(.)g(A)h(v)m(ariet)o(y)g Fn(V)25 b Fr(b)q(ecomes)0 1786 y(a)17 b(recip)q(e)799 1857 y Fn(L)d Fm(7!)g Fn(V)d Fr(\()p Fn(L)p Fr(\))j Fm(\032)g Fn(L)1124 1836 y Fk(n)0 1962 y Fr(for)20 b(pro)q(ducing)f(a)i(zero)f (set)g(giv)o(en)g(an)o(y)g Fn(L=k)r Fr(.)g(A)h(morphism)d(of)j Fm(f)p Fn(L)f Fm(7!)g Fn(V)11 b Fr(\()p Fn(L)p Fr(\))p Fm(g)21 b Fr(to)g Fm(f)p Fn(L)1694 1944 y Fj(0)1728 1962 y Fm(7!)f Fn(V)11 b Fr(\()p Fn(L)1891 1944 y Fj(0)1906 1962 y Fr(\))p Fm(g)0 2032 y Fr(w)o(ould)k(consist)h(of)h(maps)f Fn(f)k Fr(:)14 b Fn(L)g Fm(!)g Fn(L)715 2013 y Fj(0)745 2032 y Fr(and)i Fn(g)g Fr(:)e Fn(V)d Fr(\()p Fn(L)p Fr(\))k Fm(!)f Fn(V)d Fr(\()p Fn(L)1194 2013 y Fj(0)1208 2032 y Fr(\))17 b(suc)o(h)f(that)h Fn(f)5 b Fr(\()p Fn(L)p Fr(\))15 b Fm(7!)f Fn(g)r Fr(\()p Fn(V)d Fr(\()p Fn(L)p Fr(\)\).)18 b(This)0 2101 y(is)e(the)h(basis)e(for)h(sc)o(heme)g (theory)l(.)100 2171 y(F)l(or)g(general)g Fn(k)r Fr(,)h(one)g(often)g (requires)f(the)i(v)m(ariet)o(y)f Fn(V)28 b Fr(not)18 b(just)f(to)g(b)q(e)h(irreducible)d(\(so)i Fn(A)p Fr(\()p Fn(V)12 b Fr(\))18 b(is)0 2243 y(a)f(domain\))e(but)h Fp(absolutely)21 b(irreducible)p Fr(,)e(so)d(that)h(in)f(fact)p 1226 2203 V 17 w Fn(k)q Fm(\012)1292 2250 y Fk(k)1317 2243 y Fn(A)h Fr(is)f(a)g(domain.)0 2353 y Fp(Example:)k Fr(Consider)f(the)i(curv)o(e)f Fn(x)721 2335 y Fl(2)758 2353 y Fr(+)13 b Fn(y)836 2335 y Fl(2)880 2353 y Fr(=)20 b(0.)h(Then)f Fn(A)h Fr(=)f Fp(Q)p Fr([)p Fn(X)q(;)8 b(Y)k Fr(])p Fn(=)p Fr(\()p Fn(x)1498 2335 y Fl(2)1535 2353 y Fr(+)h Fn(y)1613 2335 y Fl(2)1636 2353 y Fr(\))21 b(is)f(a)h(domain,)0 2423 y(while)p 218 2453 44 2 v 218 2493 a Fp(Q)11 b Fm(\012)311 2500 y Fg(Q)358 2493 y Fn(A)k Fr(=)p 462 2453 V 13 w Fp(Q)p Fr([)p Fn(X)q(;)8 b(Y)k Fr(])p Fn(=)p Fr(\()p Fn(x)710 2472 y Fl(2)744 2493 y Fr(+)e Fn(y)819 2472 y Fl(2)842 2493 y Fr(\))875 2480 y Fm(\030)875 2495 y Fr(=)p 928 2453 V 928 2493 a Fp(Q)p Fr([)p Fn(X)q(;)e(Y)j Fr(])p Fn(=)p Fr(\()p Fn(x)h Fr(+)f Fn(y)r(i)p Fr(\))g Fm(\002)p 1360 2453 V 11 w Fp(Q)p Fr([)p Fn(X)q(;)d(Y)k Fr(])p Fn(=)p Fr(\()p Fn(x)f Fm(\000)g Fn(y)r(i)p Fr(\))0 2598 y(is)j(not.)g(One)g(could)f(require)g Fn(L)p Fm(\012)619 2605 y Fk(k)644 2598 y Fn(A)h Fr(to)h(b)q(e)f(a)g (domain)f(for)h(all)g Fn(L=k)r Fr(,)f(but)h(it)h(is)e(an)h(in)o (teresting)f(exercise)0 2670 y(in)j(comm)o(utativ)o(e)g(algebra)f(to)i (sho)o(w)e(that)p 822 2629 28 2 v 17 w Fn(k)j Fr(su\016ces.)963 2790 y(5)p eop %%Page: 6 6 6 5 bop 0 50 a Fp(2.)19 b(Dimension:)d Fr(W)l(e)h(ha)o(v)o(e)e(a)i (couple)f(of)h(p)q(ossibilities)e(for)h(de\014ning)f(dimension.)100 120 y Fq(Suggestion)i(1.)g Fr(W)l(e)f(can)f(use)h(the)g(transcendence)f (degree)g Fn(tr)o(:deg)r(:)1366 127 y Fk(k)1390 120 y Fn(Q)p Fr(\()p Fn(A)p Fr(\))j(of)e(the)g(quotien)o(t)g(\014eld)0 190 y(of)h Fn(A)g Fr(o)o(v)o(er)e Fn(k)r Fr(.)100 260
y Fq(Suggestion)20 b(2.)h Fr(Do)e(it)h(the)g(sc)o(heme-theoretic)e(w)o (a)o(y:)h(lo)q(ok)h(at)g(the)g(sp)q(ectrum)e(of)i Fn(A)p Fr(.)g(Then)f(the)0 329 y(dimension)14 b(is)h(de\014ned)g(in)g(terms)g (of)h(the)g(longest)f(sequence)h(of)g(irreducible)d(subsets)i(of)h Fn(S)s(pec)p Fr(\()p Fn(A)p Fr(\))g(\(in)0 399 y(the)g(Zariski)e(top)q (ology\).)i(This)f(is)g(what)g(is)g(kno)o(wn)g(as)g(the)g Fp(Krull)k(dimension)c Fr(of)g(a)h(ring)e Fn(A)p Fr(,)i(i.e.)f(the)0 469 y(suprem)o(um)g(o)o(v)o(er)h Fn(n)g Fm(2)f Fp(Z)j Fr(for)f(whic)o(h)g(there)g(exist)h(prime)f(ideals)f Ff(p)1273 476 y Fk(i)1307 469 y Fr(with)i Ff(p)1447 476 y Fl(0)1485 469 y Fm(\032)d Ff(p)1564 476 y Fl(1)1602 469 y Fm(\032)g(\001)8 b(\001)g(\001)16 b(\032)f Ff(p)1809 476 y Fk(n)1854 469 y Fr(in)i Fn(A)0 539 y Fr(\(the)g(con)o(tainmen)o (ts)e(here)i(are)f(strict\).)h(Note:)g(It)h(is)e(p)q(ossible)g(to)h (construct)f(No)q(etherian)h(rings)e(with)0 608 y(in\014nite)h(Krull)f (dimension.)100 679 y(W)l(e)j(will)f(stic)o(k)h(to)h(the)f(\014rst)f (de\014nition,)g(and)h(to)g(those)g(ob)s(jects)g(with)g(dimension)e(1.) h(The)h(fact)0 748 y(that)k Fn(dim)h Fm(\025)g Fr(1)f(means)e(that)j Fn(A)f Fr(has)f(a)h(nonzero)f(prime)g(ideal,)g(so)h(that)g Fn(A)h Fr(is)e(not)h(a)g(\014eld.)f(That)0 818 y Fn(dim)14 b Fm(\024)f Fr(1)k(implies)e(that)i(an)o(y)f(nonzero)f(prime)h(ideal)g (is)g(maximal.)0 928 y Fp(Example:)p 249 887 28 2 v 15 w Fn(k)10 b Fm(\012)324 935 y Fk(k)357 928 y Fn(k)r Fr([)p Fn(X)t Fr(])j(=)p 524 887 V 14 w Fn(k)q Fr([)p Fn(X)t Fr(])j(with)f(no)g(relations.)f(F)l(or)g(those)h(that)h(lik)o(e)f (relations,)f(w)o(e)h(could)g(just)0 998 y(as)h(easily)g(ha)o(v)o(e)g (used)g Fn(k)r Fr([)p Fn(X)q(;)8 b(Y)j Fr(])p Fn(=)p Fr(\()p Fn(Y)g Fr(\))17 b(in)f(place)g(of)h Fn(k)r Fr([)p Fn(X)t Fr(].)100 1068 y(The)f(w)o(ord)f(\\curv)o(e")h(will)g(alw)o(a)o (ys)g(refer)g(to)h(the)f(case)h(when)f Fn(dim)e Fr(=)f(1.)100 1138 y(If)j Fn(dim)p Fr(\()p Fn(A)p Fr(\))e(=)g(0)i(then)g Fn(A)g Fr(is)f(a)h(\014eld)f(con)o(taining)g Fn(k)r Fr(.)g(No)o(w)g Fn(k)h Fm(\032)d Fn(A)k Fr(is)e(a)h(\014eld)f(extension)g(that)i(as)e (a)0 1208 y(ring)f(is)h(generated)f(b)o(y)h(some)f(\014nite)h(set)h(o)o (v)o(er)e Fn(k)r Fr(,)g(whic)o(h)g(happ)q(ens)g(only)h(when)g([)p Fn(A)f Fr(:)f Fn(k)r Fr(])i(is)g(\014nite.)f(Keep)0 1277 y(in)h(mind)f(that)i(taking)g(recipro)q(cals)e(is)h(not)h(allo)o(w)o (ed,)e(since)h(w)o(e're)g(generating)f Fn(A)i Fr(as)f(a)h(ring,)e(not)i (as)f(a)0 1347 y(\014eld.)h(But)h(if)f(w)o(e)h(w)o(an)o(t)p 461 1306 V 15 w Fn(k)c Fm(\012)539 1354 y Fk(k)574 1347 y Fn(A)k Fr(to)g(b)q(e)g(a)g(domain,)d(then)j(of)g(necessit)o(y)l(,)e Fn(k)h Fr(=)d Fn(A)k Fr(in)f(this)h(case.)0 1457 y Fp(Example:)h Fn(k)g Fr(=)e Fp(Q)p Fr(,)i Fn(A)g Fr(=)e Fp(Q)p Fr(\()p Fn(i)p Fr(\))i(=)e Fp(Q)p Fr([)p Fn(x)p Fr(])p Fn(=)p Fr(\()p Fn(x)878 1439 y Fl(2)914 1457 y Fr(+)11 b(1\).)19 b(Then)f Fn(i)12 b Fm(\012)g Fr(1)g Fm(\000)g Fr(1)g Fm(\012)g Fn(i)19 b Fr(is)e(a)i(zero)f(divisor,)f(since)0 1527 y(\()p Fn(i)12 b Fm(\012)e Fr(1)h Fm(\000)g Fr(1)g Fm(\012)g Fn(i)p Fr(\)\()p Fn(i)h Fm(\012)f Fr(1)g(+)g(1)g Fm(\012)g Fn(i)p Fr(\))460 1597 y(=)i(\()p Fn(i)548 1579 y Fl(2)582 1597 y Fm(\012)e Fr(1\))g Fm(\000)g Fr(\(1)g Fm(\012)g Fn(i)859 1579 y Fl(2)882 1597 y Fr(\))460 1667 y(=)i(\(\()p Fm(\000)p Fr(1\))f Fm(\012)f Fr(1\))g Fm(\000)g Fr(\(1)g Fm(\012)g Fr(\()p Fm(\000)p Fr(1\)\))460 1737 y(=)i Fm(\000)p Fr(\(1)e Fm(\012)g Fr(1\))g(+)g(\(1)h Fm(\012)e Fr(1\))460 1807 y(=)j(0.)0 1877 y(More)j(generally)l(,)p 370 1968 44 2 v 370 2008 a Fp(Q)11 b Fm(\012)463 2015 y Fg(Q)510 2008 y Fn(A)k Fr(=)p 614 1968 V 13 w Fp(Q)p Fr([)p Fn(x)p Fr(])p Fn(=)p Fr(\()p Fn(x)785 1987 y Fl(2)820
2008 y Fr(+)10 b(1\))928 1994 y Fm(\030)928 2010 y Fr(=)p 980 1968 V 980 2008 a Fp(Q)p Fr([)p Fn(x)p Fr(])p Fn(=)p Fr(\()p Fn(x)i Fr(+)f Fn(i)p Fr(\))h Fm(\002)p 1310 1968 V 10 w Fp(Q)p Fr([)p Fn(x)p Fr(])p Fn(=)p Fr(\()p Fn(x)g Fm(\000)f Fn(i)p Fr(\))0 2138 y(in)16 b(this)g(example.)0 2248 y Fp(Example:)24 b Fn(y)284 2230 y Fl(2)334 2248 y Fr(=)j Fn(x)428 2230 y Fl(3)467 2248 y Fm(\000)16 b Fn(x)p Fr(.)25 b(Then)g Fn(A)i Fr(=)g Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])p Fn(=)p Fr(\()p Fn(y)1060 2230 y Fl(2)1100 2248 y Fm(\000)16 b Fn(x)1183 2230 y Fl(3)1222 2248 y Fr(+)g Fn(x)p Fr(\).)26 b(Note)f(that)g(this)f(example)h(is)0 2322 y(somewhat)17 b(analogous)f(to)h Fp(Z)p Fr([)575 2280 y Fm(p)p 617 2280 26 2 v 42 x Fn(d)p Fr(],)g(with)g Fn(d)h Fr(squarefree.)e(In)h(the)h(presen)o(t)e(case)i(w)o(e)f(ha)o(v)o (e)g Fn(k)r Fr([)p Fn(x)p Fr(][)1815 2291 y Fm(p)p 1856 2291 27 2 v 1856 2322 a Fn(y)r Fr(])e(=)0 2399 y Fn(k)r Fr([)p Fn(x)p Fr(][)98 2358 y Fm(p)p 139 2358 141 2 v 139 2399 a Fn(x)167 2385 y Fl(3)201 2399 y Fm(\000)c Fn(x)p Fr(].)19 b(When)g(w)o(e)g(dra)o(w)e(the)j(\\picture")e(of)h(the) h(curv)o(e,)e(w)o(e)h(frequen)o(tly)g(use)f(a\016ne)h(space,)0 2469 y(but)d(it)h(is)e(usually)h(nicer)f(to)i(add)f(a)g(p)q(oin)o(t)g (at)g(in\014nit)o(y)l(.)f(In)h(the)h(case)f(of)h(this)e(curv)o(e)h(o)o (v)o(er)g Fp(C)p Fr(,)g(w)o(e)g(get)h(a)0 2539 y(torus.)f(F)l(or)f (elliptic)h(curv)o(es,)g(this)g(allo)o(ws)f(us)h(to)h(mak)o(e)f(the)h (curv)o(e)f(in)o(to)g(a)g(group.)0 2670 y Fp(Lecture)k(3)c Fr(\(Septem)o(b)q(er)g(1,)h(1995\).)963 2790 y(6)p eop %%Page: 7 7 7 6 bop 0 50 a Fr(Scrib)q(e:)16 b(La)o(wren)f(Smitline)100 181 y(Recall:)h Fn(k)i Fr(is)e Fq(any)h Fr(\014eld.)100 251 y(Last)f(time,)h(w)o(e)f(sa)o(w)g(the)g(corresp)q(ondence)f(b)q(et) o(w)o(een)0 321 y Fm(f)p Fr(\(absolutely)h(irreducible\))f(a\016ne)h(v) m(arieties)g(o)o(v)o(er)g Fn(k)r Fm(g)g Fr(and)g Fm(f)p Fr(domains)f Fn(A)i Fr(of)f(\014nite)h(t)o(yp)q(e)g(o)o(v)o(er)e Fn(k)r Fm(g)p Fr(.)0 391 y(F)l(or)g(suc)o(h)h Fn(A)p Fr(,)271 378 y(\026)270 391 y Fn(k)c Fm(\012)347 398 y Fk(k)382 391 y Fn(A)17 b Fr(is)f(also)g(a)h(domain.)100 461 y(The)f(dimension)f(of)h(a)h(v)m(ariet)o(y)g(is)f(1)g(if)h(and)f (only)g(if)h(the)f(transcendence)g(degree)g(of)h Fn(Q)p Fr(\()p Fn(A)p Fr(\))h(is)e(1.)0 570 y Fp(In)n(tuition)21 b(ab)r(out)d(gluing)i(a\016ne)e(v)m(arieties)j(together:)100 681 y Fr(An)15 b(imp)q(ortan)o(t)g(c)o(haracteristic)f(of)i(the)g (global)e(top)q(ology)i(of)g(a)f(di\013eren)o(tiable)f(manifold)h(is)g (that)0 750 y(it)i(is)f(de\014ned)f(in)i(terms)e(of)i(lo)q(cal)g (homeomorphism)o(s)c(to)k(Euclidean)f(spaces.)100 860 y Fp(P)139 842 y Fk(n)166 860 y Fr(\()p Fn(k)r Fr(\))e(=)g Fm(f)p Fr([)p Fn(x)366 867 y Fl(0)388 860 y Fn(;)8 b(:)g(:)g(:)h(;)f(x) 527 867 y Fk(n)554 860 y Fr(])14 b(:)g Fm(9)p Fn(i;)24 b(x)721 867 y Fk(i)752 860 y Fm(6)p Fr(=)14 b(0)p Fm(g)f Fr(=)h Fn(k)949 842 y Fk(n)p Fl([phone] y Fm(\000)d(f)p Fp(0)p Fm(g)p Fn(=)16 b Fr(\(scalar)g(m)o(ultiplication\).)100 970 y(F)l(or)f(example,)h Fp(P)437 952 y Fl(1)459 970 y Fr(\()p Fn(k)r Fr(\))f(=)e Fm(f)p Fr([)p Fn(x;)8 b(y)r Fr(])14 b(:)g Fn(x;)8 b(y)17 b Fm(2)d Fn(k)r(;)24 b Fr(\()p Fn(x;)8 b(y)r Fr(\))15 b Fm(6)p Fr(=)f(\(0)p Fn(;)8 b Fr(0\))p Fm(g)14 b Fr(=)g Fm(f)p Fr([)p Fn(x;)8 b Fr(1])p Fm(g)j([)g(f)p Fr([1)p Fn(;)d Fr(0])p Fm(g)p Fr(.)100 1040 y(Ho)o(w)16 b(can)g(w)o(e)h(obtain)f Fp(P)572 1022 y Fl(1)611 1040 y Fr(b)o(y)g(gluing?)0 1110 y(Geometrically)l(,)g(w)o (e)g(ha)o(v)o(e)g Fp(P)557 1092 y Fl(1)591 1110 y Fm(\000)10
b(f1g)k Fr(=)f Fp(A)849 1092 y Fl(1)886 1110 y Fm(\033)g Fp(A)981 1092 y Fl(1)1015 1110 y Fm(\000)e(f)p Fr(0)p Fm(g)j(\032)f Fp(A)1249 1092 y Fl(1)1283 1110 y Fm([)e(f1g)g(\000)g(f)p Fr(0)p Fm(g)i Fr(=)h Fp(P)1668 1092 y Fl(1)1701 1110 y Fm(\000)d(f)p Fr(0)p Fm(g)p Fr(.)0 1180 y(Algebraically)l(,)k Fn(k)r Fr([)p Fn(x)p Fr(])f Fm(\032)g Fn(k)r Fr([)p Fn(x;)8 b(x)582 1162 y Fj(\000)p Fl(1)635 1180 y Fr(])14 b Fm(\033)g Fn(k)r Fr([)p Fn(x)786 1162 y Fj(\000)p Fl(1)839 1180 y Fr(].)100 1290 y(No)o(w)g(w)o(e)g(ha)o(v)o(e)f(to)i(\014x)f(a)g (problem.)e(En)o(tire)i(functions)f(on)h(compact)g(v)m(arieties)g(are)g (constan)o(t.)f(F)l(or)0 1360 y(example,)j(maps)f(from)h Fp(C)11 b Fm([)h(f1g)h(!)h Fp(C)j Fr(are)f(constan)o(t.)100 1470 y(W)l(e)g(also)g(don't)g(w)o(an)o(t)g(to)h(see)g(the)f(\\scars")g (from)f(the)i(gluing.)100 1580 y(Notice)24 b(that)f(all)g(the)g(rings)f (w)o(e)h(w)o(an)o(t)f(to)i(glue)f(together)g(corresp)q(ond)e(to)j(co)q (ordinate)e(rings)0 1650 y(giving)16 b(the)h(same)f(\014eld)g(of)g (fractions.)0 1720 y(W)l(e)h(pic)o(k)f(a)g(p)q(oin)o(t,)g Fn(P)24 b Fr(on)16 b(eac)o(h)g(piece)h Fp(A)781 1701 y Fl(1)814 1720 y Fm([)12 b(f1g)e(\000)h(f)p Fr(0)p Fm(g)p Fn(;)25 b Fp(A)1176 1701 y Fl(1)1198 1720 y Fn(;)8 b Fp(A)1263 1701 y Fl(1)1297 1720 y Fm([)j(f1g)p Fn(:)0 1789 y Fr(W)o(OLOG,)16 b Fn(P)21 b Fr(=)13 b(3.)k(W)l(e)f(ha)o(v)o(e)g (3)e(:)g Fn(k)r Fr([)p Fn(x)p Fr(])f Fm(!)h Fn(k)r Fr(,)i(with)g(k)o (ernel)g(\()p Fn(x)c Fm(\000)f Fr(3\))17 b(and)e(also)h(the)0 1859 y(map)g(3)d(:)h Fn(k)r Fr([)p Fn(x)247 1841 y Fj(\000)p Fl(1)300 1859 y Fr(])g Fm(!)g Fn(k)r Fr(,)i(with)g(k)o(ernel)g(\()p Fn(x)757 1841 y Fj(\000)p Fl(1)822 1859 y Fm(\000)878 1840 y Fl(1)p 878 1848 20 2 v 878 1877 a(3)904 1859 y Fr(\).)100 1970 y(The)g(lo)q(cal)h(rings)e(on)h(the)h(a\016ne)f(pieces) g(agree.)100 2040 y(A)g Fp(lo)r(cal)j(ring)d Fr(is)f(a)h(ring)e Fn(R)i Fr(suc)o(h)f(that)h Fn(R)9 b Fm(\000)g Fn(R)999 2021 y Fj(\002)1049 2040 y Fr(is)15 b(an)g(ideal.)g Fn(R)h Fr(is)f(lo)q(cal)h(if)g Fn(R)g Fr(has)f(exactly)h(one)0 2109 y(maximal)f(ideal.)100 2219 y(If)h Fn(A)h Fr(is)f(a)h(ring)f(and)f Ff(p)i Fr(is)f(a)h(prime)e(ideal,)h(then)g(let)h Fn(A)1124 2226 y Fe(p)1161 2219 y Fr(b)q(e)g(the)g(ring)e(\()p Fn(A)d Fm(\000)f Ff(p)p Fr(\))1579 2201 y Fj(\000)p Fl(1)1632 2219 y Fn(A)p Fr(.)100 2350 y Fp(A)143 2332 y Fk(n)170 2350 y Fr(\()p Fn(k)r Fr(\))21 b(=)g Fn(k)345 2332 y Fk(n)392 2350 y Fm(\032)g Fp(P)491 2332 y Fk(n)518 2350 y Fr(\()p Fn(k)r Fr(\),)g(b)o(y)f(\()p Fn(x)739 2357 y Fl(1)762 2350 y Fn(;)8 b(:)g(:)g(:)h(;)f(x)901 2357 y Fk(n)928 2350 y Fr(\))22 b Fm(7!)e Fr(\(1)p Fn(;)8 b(x)1133 2357 y Fl(1)1156 2350 y Fn(;)g(:)g(:)g(:)h(;)f(x)1295 2357 y Fk(n)1323 2350 y Fr(\).)21 b(In)f(the)h(study)f(of)h(pro)s (jectiv)o(e)0 2420 y(v)m(arieties,)16 b(w)o(e)h(examine)f(only)g(zero)g (sets)h(of)g(homogeneous)d(p)q(olynomials.)100 2530 y(A)j(v)m(ariet)o (y)f(is)g(an)h(absolutely)f(irreducible)e(in)o(tegral)i(separated)f(sc) o(heme)h(of)g(\014nite)h(t)o(yp)q(e)g(o)o(v)o(er)e Fn(k)r Fr(.)0 2600 y Fq(Question:)i Fr(Is)g(a)f(v)m(ariet)o(y)h(o)o(v)o(er)f Fn(k)i Fr(alw)o(a)o(ys)d(quasipro)s(jectiv)o(e?)0 2670 y Fq(Question:)i Fr(Giv)o(en)g(a)f(function)g(\014eld,)g(is)g(there)h (a)f(curv)o(e)g(y)o(ou)g(can)g(deduce?)963 2790 y(7)p eop %%Page: 8 8 8 7 bop 100 50 a Fr(Supp)q(ose)22 b Fn(k)28 b Fr(=)422 37 y(\026)421 50 y Fn(k)q Fr(.)c(Then)f(there)h(is)g(a)f(\(con)o(tra)o (v)m(arian)o(t)g(functor?\))g(corresp)q(ondence)f(b)q(et)o(w)o(een)0 120 y(smo)q(oth)c(pro)s(jectiv)o(e)f(curv)o(es)h(o)o(v)o(er)f
Fn(k)j Fr(and)d(\014nitely)h(generated)g(\014eld)f(extensions)h Fn(K)k Fr(o)o(v)o(er)17 b Fn(k)j Fr(of)e(tran-)0 189 y(scendence)e(degree)g(1.)100 302 y Fo(Problems)e(with)j(smo)q(othness) e Fr(when)h(one)g(has)g(singularities:)100 415 y(F)l(or)f(example,)h (the)h(curv)o(e)f Fn(y)643 397 y Fl(2)679 415 y Fr(=)e Fn(x)760 397 y Fl(3)794 415 y Fr(+)c Fn(x)871 397 y Fl(2)911 415 y Fr(has)16 b(a)g(cusp,)g(or)g(at)h(least)f(the)h(real)f(part)g(do) q(es.)100 489 y(If)21 b Fn(k)i Fr(=)264 475 y(\026)263 489 y Fn(k)q Fr(,)e(then)g(a)g(p)q(oin)o(t)g Fn(P)28 b Fr(on)20 b(a)h(curv)o(e)g(is)f(smo)q(oth)h(i\013)g(the)g(lo)q(cal)g (ring)f(at)i Fn(P)28 b Fr(is)20 b(a)h Fq(discr)m(ete)0 558 y(valuation)d(ring)p Fr(.)100 630 y(T)l(une)f(in)h(next)g(time)g (for)g(\\a)g(new)g(b)q(eginning".)e(W)l(e'll)i(see)g(that)h(certain)e (\014elds)g(corresp)q(ond)g(to)0 699 y(curv)o(es)f(and)f(p)q(oin)o(ts)h (on)g(the)h(curv)o(es)f(corresp)q(ond)f(to)h(lo)q(cal)h(rings.)0 838 y Fp(Lecture)j(4)c Fr(\(Septem)o(b)q(er)g(6,)h(1995\).)0 909 y(Scrib)q(e:)f(Nic)o(k)h(Mey)o(er)100 1049 y(T)l(o)q(da)o(y)g(w)o (e)g(will)h(b)q(egin)f(exploring)g(the)h(corresp)q(ondence)e(b)q(et)o (w)o(een)h(certain)g(\014elds)g(and)g(curv)o(es;)0 1119 y(w)o(e)25 b(will)f(\014nd)g(that)h(p)q(oin)o(ts)f(on)g(the)h(curv)o(e) f(corresp)q(ond)f(to)i(lo)q(cal)g(rings)f(of)h(the)g(\014eld.)f(In)g (short,)0 1188 y Fn(P)d Fm(2)14 b Fn(C)j Fm( )-8 b(!)13 b(O)298 1195 y Fk(P)345 1188 y Fm(\032)g Fn(K)t Fr(\()p Fn(C)t Fr(\))h(=)f Fn(K)t Fr(.)100 1260 y Fp(V)-5 b(aluation)20 b(rings)p Fr(:)100 1331 y Fp(De\014nition)p Fr(:)d(Let)h Fn(K)i Fr(b)q(e)d(a)f(\014eld.)g(A)h Fp(v)m(aluation)j(ring)f(of)g Fn(K)h Fr(is)c(a)h(subring)d Fn(R)g Fm(\032)g Fn(K)20 b Fr(suc)o(h)c(that)0 1401 y Fm(8)p Fn(x)d Fm(2)h Fn(K)d Fm(\000)c Fn(R;)h(x)304 1383 y Fj(\000)p Fl(1)371 1401 y Fm(2)14 b Fn(R)p Fr(.)g(A)h Fp(v)m(aluation)j(ring)e(of)g Fn(K)q(=k)g Fr(is)e(a)g(v)m(aluation)g(ring)f(of)i Fn(K)j Fr(suc)o(h)13 b(that)i Fn(k)g Fm(\032)f Fn(R)p Fr(.)0 1471 y(Finally)l(,)h(a)i Fp(v)m(aluation)j(ring)d Fr(is)f(a)g(domain)f (that)i(is)f(a)h(v)m(aluation)f(ring)g(of)g(its)h(\014eld)f(of)g (fractions.)100 1542 y Fp(Example)p Fr(:)g Fn(R)e Fr(=)f Fn(K)t Fr(.)j(The)h(\\trivial")f(v)m(aluation)g(ring.)100 1614 y Fp(De\014nition)p Fr(:)k(A)f Fp(discrete)j(v)m(aluation)h(ring)c Fr(of)g Fn(K)j Fr(is)d(a)f(v)m(aluation)h(ring)e Fn(R)i Fr(of)g Fn(K)k Fr(for)18 b(whic)o(h)0 1683 y Fn(K)46 1665 y Fj(\003)69 1683 y Fn(=R)132 1665 y Fj(\003)171 1683 y Fr(is)e(in\014nite)g(cyclic)h(as)f(a)h(group.)100 1755 y Fp(Example)p Fr(:)i Fn(k)i Fm(\032)f Fn(R)g Fr(=)f Fn(k)r Fr([[)p Fn(t)p Fr(]])g Fm(\032)g Fn(K)k Fr(=)d Fn(k)r Fr(\(\()p Fn(t)p Fr(\)\),)h(where)e Fn(k)r Fr([[)p Fn(t)p Fr(]])g(and)h Fn(k)r Fr(\(\()p Fn(t)p Fr(\)\))h(are)f(the)g(p)q (o)o(w)o(er)f(and)0 1825 y(Lauren)o(t)e(series)g(rings)f(in)h Fn(t)p Fr(,)h(resp)q(ectiv)o(ely)l(.)g(Note)g(that)g Fn(R)1107 1806 y Fj(\003)1146 1825 y Fr(=)e Fm(f)p Fn(f)22 b Fm(2)16 b Fn(R)g Fm(j)g Fn(f)5 b Fr(\(0\))17 b Fm(6)p Fr(=)f(0)p Fm(g)p Fr(;)h Fn(K)1696 1806 y Fj(\003)1735 1825 y Fr(=)e Fn(t)1807 1806 y Fg(Z)1849 1825 y Fm(\001)d Fn(R)1913 1806 y Fj(\003)1936 1825 y Fr(;)0 1894 y Fn(R)f Fm(\000)g(f)p Fr(0)p Fm(g)j Fr(=)f Fn(t)258 1876 y Fg(Z)p Fj(\025)p Fl(0)350 1894 y Fm(\001)e Fn(R)413 1876 y Fj(\003)436 1894 y Fr(.)100 1966 y(It)g(turns)f(out)h(that)g Fn(K)504 1948 y Fj(\003)527 1966 y Fn(=R)590 1948 y Fj(\003)624 1966 y Fr(is)g(a)g(totally)g(ordered)f(ab)q(elian)h(group,)e(with)i Fn(aR)1522 1948 y Fj(\003)1559 1966 y Fm(\025)j Fn(bR)1671 1948 y Fj(\003)1705 1966 y Fr(if)e Fn(aR)i Fm(\022)g
Fn(bR)p Fr(.)0 2035 y(In)i(fact,)h Fn(K)220 2017 y Fj(\003)243 2035 y Fn(=R)306 2017 y Fj(\003)346 2035 y Fr(is)f Fp(torsion-free)p Fr(.)100 2107 y Fp(Prop)r(erties)i(of)h(v)m(aluation)h(rings:)100 2178 y(Prop)r(osition)p Fr(:)f(An)o(y)g(v)m(aluation)g(ring)f Fn(R)i Fr(is)f Fp(lo)r(cal)p Fr(;)i(i.e.,)e(it)g(has)g(exactly)h(one)g (maximal)e(ideal.)0 2248 y(This)e(maximal)f(ideal)h(is,)g(of)h(course,) e Fn(R)d Fm(\000)e Fn(R)860 2230 y Fj(\003)883 2248 y Fr(.)100 2320 y Fp(Pro)r(of)p Fr(:)21 b(It)j(su\016ces)f(to)g(pro)o(v)o (e)g(that)h Fn(R)16 b Fm(\000)f Fn(R)984 2302 y Fj(\003)1031 2320 y Fr(is)23 b(an)g(ideal.)g(This)g(is)g(quite)h(easy)l(.)f(First,)f (it's)0 2389 y(nonempt)o(y)l(,)17 b(since)h(0)23 b Fn(=)-31 b Fm(2)18 b Fn(R)499 2371 y Fj(\003)522 2389 y Fr(.)h(Second,)e(if)i Fn(a)f Fm(2)f Fn(R)c Fm(\000)f Fn(R)1021 2371 y Fj(\003)1044 2389 y Fn(;)c(r)19 b Fm(2)f Fn(R)p Fr(,)g(then)h Fn(r)q(a)24 b(=)-31 b Fm(2)18 b Fn(R)1500 2371 y Fj(\003)1523 2389 y Fr(,)g(ob)o(viously)l(.)f(Finally)l(,)0 2459 y Fn(a;)8 b(b)15 b Fm(2)f Fn(R)c Fm(\000)h Fn(R)267 2441 y Fj(\003)304 2459 y Fm(\))i Fn(a)e Fr(+)f Fn(b)k Fm(2)h Fn(R)10 b Fm(\000)g Fn(R)671 2441 y Fj(\003)695 2459 y Fr(.)16 b(\(Pro)q(of:)g Fn(a)11 b Fr(+)f Fn(b)k Fr(=)g Fn(a)p Fr(\(1)d(+)f Fn(b=a)p Fr(\).)17 b(If)g Fn(b=a)d Fm(2)g Fn(R)p Fr(,)j(then)f(w)o(e're)f(\014ne;)h(if)0 2529 y Fn(b=a)h Fr(is)f(not)h(in)f Fn(R)p Fr(,)h Fn(a=b)g Fr(is,)f(and)g (since)g Fn(a)11 b Fr(+)g Fn(b)j Fr(=)g Fn(b)p Fr(\(1)d(+)g Fn(a=b)p Fr(\),)18 b(w)o(e're)d(still)i(\014ne\).)100 2600 y(All)h(v)m(aluation)f(rings)g(are)h(lo)q(cal,)g(but)g(not)g(all)f (lo)q(cal)h(rings)f(are)h(v)m(aluation)f(rings;)g(for)h(example,)0 2670 y Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])132 2677 y Fk(<x;y)q(>)269 2670 y Fr(is)16 b(lo)q(cal,)h(but)f(is)g(not)h(a)f(v)m(aluation)g (ring.)963 2790 y(8)p eop %%Page: 9 9 9 8 bop 100 50 a Fp(Prop)r(osition)p Fr(:)17 b(Let)h Fn(R)g Fr(b)q(e)g(a)g(ring.)f(Then)g Fn(R)h Fr(is)f(a)h(discrete)f(v)m (aluation)h(ring)f(\(d.v.r.\))g(i\013)h Fn(R)g Fr(is)f(a)0 120 y(principal)e(ideal)h(domain)f(with,)h(up)g(to)h(m)o(ultiplication) e(b)o(y)h(units,)f(exactly)j(one)e(prime)g(elemen)o(t.)100 195 y Fp(Pro)r(of)p Fr(:)100 271 y(\()p Fm(\()p Fr(\):)j Fn(R)13 b Fm(\000)f Fr(0)18 b(=)g Fn(R)461 252 y Fj(\003)497 271 y Fm(\002)12 b Fn(\031)578 252 y Fg(Z)p Fj(\025)p Fl(0)659 271 y Fr(,)19 b(where)g Fn(\031)i Fr(is)d(our)g(prime)g (elemen)o(t.)g(Note)i(that)g(this)e(is)h(a)g(unique)0 340 y(decomp)q(osition,)i(since)i Fn(R)g Fr(is)g(a)g(p.i.d.,)e(b)o(y)i (assumption.)d(Then)j Fn(K)1332 322 y Fj(\003)1379 340 y Fr(=)h Fn(R)1480 322 y Fj(\003)1519 340 y Fm(\002)15 b Fn(\031)1603 322 y Fg(Z)1656 340 y Fr(is)23 b(the)g(unique)0 410 y(decomp)q(osition)15 b(in)h(the)h(\014eld)f(of)h(fractions.)100 486 y(\()p Fm(\))p Fr(\):)g Fn(K)265 467 y Fj(\003)288 486 y Fn(=R)351 467 y Fj(\003)390 486 y Fr(=)e(\()p Fn(\031)r(R)531 467 y Fj(\003)554 486 y Fr(\))573 467 y Fg(Z)604 486 y Fr(,)i(where)g Fn(\031)i Fr(is)e(the)h(generator)f(of)g(the)h (in\014nite)e(cyclic)i(group)e Fn(K)1827 467 y Fj(\003)1850 486 y Fn(=R)1913 467 y Fj(\003)1936 486 y Fr(.)0 555 y(W)l(e)g(can)g(c)o(ho)q(ose)g(either)g Fn(\031)i Fr(or)e Fn(\031)612 537 y Fj(\000)p Fl(1)682 555 y Fr(to)h(b)q(e)f(in)g Fn(R)p Fr(;)g(WLOG,)g(c)o(ho)q(ose)g Fn(\031)g Fm(2)e Fn(R)p Fr(.)i(Note)h(that)g Fn(\031)k(=)-30 b Fm(2)14 b Fn(R)1800 537 y Fj(\003)1823 555 y Fr(.)i(No)o(w)0 625 y Fn(\031)28 632 y Fk(n)69 625 y Fm(2)e Fn(R)28 b Fm(\()-8 b(\))27 b Fn(n)14 b Fm(\025)g Fr(0.)100 701
y(This)g(fact)i(can)e(b)q(e)i(useful.)e(F)l(or)g(example:)g(let)h Fn(I)j Fm(\032)c Fn(R)h Fr(b)q(e)g(an)o(y)g(nonzero)f(ideal.)g(Pic)o(k) h(an)g(elemen)o(t)0 770 y Fn(u)g Fm(\001)g Fn(\031)103 752 y Fk(n)154 770 y Fm(2)24 b Fn(I)i Fr(with)d Fn(n)f Fr(as)g(small)g(as)g(p)q(ossible.)f(Then)h Fn(u)15 b Fm(\001)g Fn(\031)1148 752 y Fk(n)1198 770 y Fr(divides)21 b(an)o(y)h(elemen)o(t)g(of)h Fn(I)t Fr(.)f(So)h Fn(I)k Fr(=)0 840 y Fn(R)11 b Fm(\001)g Fn(u)g Fm(\001)g Fn(\031)169 822 y Fk(n)210 840 y Fr(=)i Fn(R)f Fm(\001)f Fn(\031)367 822 y Fk(n)394 840 y Fr(.)100 916 y Fp(Example)p Fr(:)16 b Fp(Z)383 925 y Fl(\(37\))470 916 y Fr(=)d Fn(n=m)h Fr(:)f Fn(n;)8 b(m;)g Fm(2)14 b Fp(Z)p Fn(;)8 b(m)20 b(=)-31 b Fm(2)14 b Fr(37)p Fp(Z)i Fr(is)h(a)f(discrete)g(v)m(aluation) g(ring.)100 991 y Fp(Example)p Fr(:)f(Let)i Fn(k)h Fr(b)q(e)e(a)g (\014eld,)f Fn(t)i Fr(a)f(transcenden)o(tal)e(o)o(v)o(er)h Fn(k)r Fr(.)h(W)l(e)g(will)g(pro)q(ceed)g(to)g(describ)q(e)f(all)0 1061 y(v)m(aluation)h(rings)f(of)i Fn(k)r Fr(\()p Fn(t)p Fr(\))g(o)o(v)o(er)f Fn(k)r Fr(.)100 1136 y(Let)h Fn(k)e Fm(\032)f Fn(R)g Fm(\032)f Fn(k)r Fr(\()p Fn(t)p Fr(\))k(b)q(e)g(a)g(v) m(aluation)f(ring.)100 1267 y(Case)g(1:)g Fn(t)e Fm(2)g Fn(R)p Fr(.)j(Then)560 1207 y Fn(k)r Fr([)p Fn(t)p Fr(])49 b Fm(\032)h Fn(R)580 1267 y Fm([)161 b([)584 1326 y Ff(p)74 b Fm(\032)50 b Ff(m)100 1397 y Fr(where)18 b Ff(m)h Fr(is)f(the)h (maximal)e(ideal)i(of)g Fn(R)p Fr(,)f(and)g Ff(p)g Fr(=)f Fn(k)r Fr([)p Fn(t)p Fr(])11 b Fm(\\)i Ff(m)19 b Fr(is)f(a)h(prime)e (ideal)i(of)f Fn(k)r Fr([)p Fn(t)p Fr(].)g(Notice)0 1467 y(that)f Fn(k)r Fr([)p Fn(t)p Fr(])10 b Fm(\000)h Ff(p)j Fr(=)f Fn(S)k Fm(\032)c Fn(R)471 1449 y Fj(\003)508 1467 y Fm(\))h Fn(k)r Fr([)p Fn(t)p Fr(])646 1474 y Fe(p)679 1467 y Fr(=)f Fn(S)765 1449 y Fj(\000)p Fl(1)818 1467 y Fn(k)r Fr([)p Fn(t)p Fr(])g Fm(\032)h Fn(R)p Fr(.)100 1542 y(Case)i(1a:)g Ff(p)e Fr(=)g(0.)i(Then)g Fn(k)r Fr([)p Fn(t)p Fr(])650 1549 y Fe(p)683 1542 y Fr(=)d Fn(k)r Fr(\()p Fn(t)p Fr(\))i Fm(\032)e Fn(R)h Fm(\032)g Fn(k)r Fr(\()p Fn(t)p Fr(\).)100 1618 y(Case)e(1b:)g Ff(p)i Fr(=)g(\()p Fn(f)5 b Fr(\))p Fn(;)j(f)21 b Fm(2)14 b Fn(k)r Fr([)p Fn(t)p Fr(],)e Fn(f)18 b Fr(an)12 b(irreducible)f(p)q (olynomial.)h(Then)g Fn(k)r Fr([)p Fn(t)p Fr(])1485 1627 y Fl(\()p Fk(f)t Fl(\))1541 1618 y Fr(,)g(the)h(lo)q(calization)g(of)0 1688 y Fn(k)r Fr([)p Fn(t)p Fr(])f(at)g(\()p Fn(f)5 b Fr(\),)14 b(is)e(a)g(d.v.r.)g(con)o(tained)f(in)h Fn(R)p Fr(.)h(So)f Fn(k)r Fr(\()p Fn(t)p Fr(\))937 1670 y Fj(\003)960 1688 y Fn(=k)r Fr([)p Fn(t)p Fr(])1059 1670 y Fj(\003)1059 1704 y Fl(\()p Fk(f)t Fl(\))1127 1688 y Fr(is)g(in\014nite)g(cyclic,)g (and)g(maps)f(surjectiv)o(ely)0 1765 y(on)o(to)18 b Fn(k)r Fr(\()p Fn(t)p Fr(\))198 1747 y Fj(\003)221 1765 y Fn(=R)284 1747 y Fj(\003)307 1765 y Fr(.)g(The)g(torsion-freeness)d(of)k Fn(k)r Fr(\()p Fn(t)p Fr(\))938 1747 y Fj(\003)961 1765 y Fn(=R)1024 1747 y Fj(\003)1065 1765 y Fr(means)e(that)h(this)g(map)f (is)h(an)f(isomorphism;)0 1834 y(therefore)f Fn(k)r Fr([)p Fn(t)p Fr(])283 1843 y Fl(\()p Fk(f)t Fl(\))353 1834 y Fr(=)e Fn(R)p Fr(.)100 1970 y(Case)i(2:)g Fn(t)k(=)-31 b Fm(2)14 b Fn(R)p Fr(.)j(Then)f Fn(u)10 b Fm(\000)h Fr(1)p Fn(=t)j Fm(2)g Fn(R)d Fm(\000)g Fn(R)907 1952 y Fj(\003)930 1970 y Fr(.)17 b(W)l(e)f(ha)o(v)o(e)1169 1910 y Fn(k)r Fr([)p Fn(u)p Fr(])49 b Fm(\032)h Fn(R)1195 1970 y Fm([)166 b([)1178 2030 y Fr(\()p Fn(u)p Fr(\))58 b Fm(\032)50 b Ff(m)1438 1970 y Fr(.)100 2100 y(So)19 b Fn(R)h Fr(=)g Fn(k)r Fr([)p Fn(u)p Fr(])374 2109 y
Fl(\()p Fk(u)p Fl(\))449 2100 y Fr(=)g Fm(f)p Fn(d)p Fr(\()p Fn(u)p Fr(\))p Fn(=h)p Fr(\()p Fn(u)p Fr(\))p Fm(j)p Fn(d)p Fr(\(0\))p Fn(=h)p Fr(\(0\)is)d(w)o(ell-de\014ned)n Fm(g)i Fr(=)h Fm(f)p Fn(f)5 b(=g)22 b Fr(:)d Fn(f)s(;)8 b(g)22 b Fm(2)e Fn(k)r Fr([)p Fn(t)p Fr(])p Fn(;)8 b Fr(deg)g Fn(f)25 b Fm(\024)0 2170 y Fr(deg)9 b Fn(g)r Fm(g)16 b Fr(\()e Fm(\()-8 b(\))27 b Fn(f)5 b Fr(\()p Fm(1)p Fr(\))p Fn(=g)r Fr(\()p Fm(1)p Fr(\))18 b(w)o(ell-de\014ned\).) 100 2246 y Fp(Prop)r(osition)p Fr(:)d(Ev)o(ery)i(non-trivial)d(v)m (aluation)i(ring)g Fn(R)g Fr(of)h Fn(k)r Fr(\()p Fn(t)p Fr(\))g(o)o(v)o(er)e Fn(K)20 b Fr(is)c(discrete.)g(Also,)g(the)0 2316 y(set)h(of)f(those)h Fn(R)p Fr('s)f(is)g(isomorphic)e(to)j(the)g (set)g(of)g(monic)e(irreducible)g(p)q(olynomials)g(in)h Fn(k)r Fr([)p Fn(t)p Fr(],)g(plus)f Fm(1)p Fr(.)100 2391 y(W)l(e)23 b(no)o(w)f(quic)o(kly)h(explain)f(wh)o(y)g(these)h(ob)s (jects)g(are)f(called)h(v)m(aluation)f(rings.)f(The)i(natural)0 2461 y(mapping)c Fn(K)255 2443 y Fj(\003)298 2461 y Fm(!)h Fn(K)414 2443 y Fj(\003)437 2461 y Fn(=R)500 2443 y Fj(\003)544 2461 y Fm(\021)g Fp(Z)g Fr(induces)g(a)g(map)g Fn(v)i Fr(:)f Fn(R)14 b Fm(\000)f Fr(0)20 b Fm(!)h Fp(Z)f Fm(\025)g Fr(0.)h(\()p Fn(v)r Fr(\(0\))g(is)f(de\014ned)g(to)h(b)q(e)0 2531 y(+)p Fm(1)p Fr(\).)d(It)h(can)f(b)q(e)g(easily)g(seen)g(that)h Fn(v)r Fr(\()p Fn(a)12 b Fr(+)g Fn(b)p Fr(\))18 b Fm(\025)e Fr(min)7 b Fn(v)r Fr(\()p Fn(a)p Fr(\))p Fn(;)h(v)r Fr(\()p Fn(b)p Fr(\))s(.)18 b(In)g(the)g(case)g(where)g(the)h(d.v.r.)e Fn(R)0 2600 y Fr(corresp)q(onds)d(to)i(an)f(irreducible)f(p)q (olynomial)h Fn(f)5 b Fr(,)17 b Fn(v)r Fr(\()p Fn(r)f Fm(2)e Fn(R)p Fr(\))i(coun)o(ts)f(the)h(n)o(um)o(b)q(er)e(of)i(factors) g(of)g Fn(f)21 b Fr(in)0 2670 y Fn(r)q Fr(.)c(Where)g Fn(r)h Fr(corresp)q(onds)c(to)j(the)g(p)q(oin)o(t)f("at)h(in\014nit)o (y",)e Fn(v)r Fr(\()p Fn(g)r(=h)p Fr(\))f(=)g(deg)8 b Fn(h)j Fm(\000)g Fr(deg)e Fn(g)r Fr(.)963 2790 y(9)p eop %%Page: 10 10 10 9 bop 0 50 a Fp(Lecture)20 b(5)c Fr(\(Septem)o(b)q(er)g(8,)h (1995\).)0 120 y(Scrib)q(e:)f(Matt)h(Bak)o(er)100 237 y(T)l(o)q(da)o(y)f(w)o(e're)g(going)f(to)i(pro)o(v)o(e)f(some)g (theorems.)100 346 y Fp(Recall)p Fr(:)i Fn(K)g Fm(\033)c Fn(R)j Fr(v)m(aluation)f(ring.)100 456 y Fp(Problem:)j Fr(Giv)o(en)g Fn(K)k Fm(\032)d Fn(L)f Fr(an)h(extension)f(of)i (\014elds,)d Fn(R)i Fm(\032)f Fn(K)24 b Fr(a)c(v)m(aluation)f(ring,)g (is)g(there)h(a)0 526 y(v)m(aluation)c(ring)g Fn(T)23 b Fr(of)17 b Fn(L)g Fr(suc)o(h)e(that)i Fn(T)h Fm(\\)11 b Fn(K)18 b Fr(=)c Fn(R)p Fr(?)100 635 y(It)h(turns)f(out)h(that)g(suc) o(h)f Fn(T)22 b Fr(exists,)15 b(and)g(w)o(e)f(will)h(try)g(to)h(c)o (haracterize)e(what)h(prop)q(erties)f(it)h(has.)100 745 y Fp(Theorem)20 b(1)p Fr(:)f(If)h Fn(K)k Fr(is)19 b(a)g(\014eld)g(with) g(v)m(aluation)h(ring)e Fn(R)p Fr(,)i(and)e Fn(L)h Fm(\033)g Fn(K)k Fr(is)c(a)h(\014eld)e(extension,)0 815 y(then)e Fm(9)h Fr(a)f(v)m(aluation)g(ring)g Fn(T)24 b Fr(of)16 b Fn(L)h Fr(with)f Fn(T)i Fm(\\)12 b Fn(K)17 b Fr(=)d Fn(R)p Fr(.)100 924 y(F)l(or)g(the)h(pro)q(of)g(of)g(this,)g(let)g Fn(K)k Fr(b)q(e)d(a)f(\014eld,)f(\012)i(an)e(algebraically)g(closed)h (\014eld,)f(and)h(let)g(\006)g(b)q(e)h(the)0 994 y(set)h(of)g(all)f (pairs)f(\()p Fn(B)r(;)8 b(g)r Fr(\),)17 b(where)f Fn(B)j Fr(is)d(a)h(subring)d(of)j Fn(K)j Fr(and)c Fn(g)g Fr(:)d Fn(B)j Fm(!)e Fr(\012)j(is)f(a)g(ring)g(homomorphism)o(.)0 1064 y(Since)h(this)g(will)g(all)g(b)q(e)h(rather)e(v)m(acuous)h(if)h (\006)d(=)g Fm(;)p Fr(,)i(one)g(can)g(c)o(hec)o(k)h(as)f(an)g(exercise)
g(that)h(\006)d(=)g Fm(;)g(,)0 1134 y Fr(c)o(har)20 b(\012)h Fm(6)p Fr(=)g(c)o(har)f Fn(K)25 b(>)c Fr(0.)g(W)l(e)g(put)g(a)g (partial)f(order)g(on)g(\006)h(b)o(y)g(sa)o(ying)f(that)h(\()p Fn(B)r(;)8 b(g)r Fr(\))22 b Fm(\024)f Fr(\()p Fn(B)1804 1115 y Fj(0)1819 1134 y Fn(;)8 b(g)1867 1115 y Fj(0)1881 1134 y Fr(\))21 b(if)0 1203 y Fn(B)26 b Fm(\032)d Fn(B)166 1185 y Fj(0)202 1203 y Fr(and)f Fn(g)331 1185 y Fj(0)368 1203 y Fm(j)382 1210 y Fk(B)416 1203 y Fr(=)h Fn(g)r Fr(.)f(If)g(\006)i Fm(6)p Fr(=)f Fm(;)p Fr(,)f(then)g(Zorn's)g(Lemma)f (tells)h(us)g(that)h(\006)f(has)f(at)i(least)f(one)0 1273 y(maximal)15 b(elemen)o(t)h(\()p Fn(T)t(;)8 b(h)p Fr(\).)18 b(No)o(w)e(w)o(e)h(apply)f(the)g(follo)o(wing)g(Theorem:)100 1383 y Fp(Theorem)g(2)p Fr(:)e(If)i(\()p Fn(B)r(;)8 b(g)r Fr(\))16 b(is)f(maximal)f(then)i Fn(B)h Fr(is)e(a)h(v)m(aluation)f (ring)f(of)i Fn(K)j Fr(with)c(maximal)g(ideal)0 1452 y(equal)h(to)h(k)o(er\()p Fn(g)r Fr(\).)100 1564 y(No)o(w)i(w)o(e)g(ha) o(v)o(e)g(a)h(map)f Fn(g)h Fr(:)f Fn(R)g Fm(!)p 771 1521 102 2 v 18 w Fn(R=)p Ff(m)h Fr(=)e(\012,)i(and)e(w)o(e)i(can)f(assume)f (that)i Fn(R)f Fm(\032)g Fn(T)26 b Fm(\032)18 b Fn(L)p Fr(,)i(and)0 1634 y Fn(h)c Fr(:)g Fn(T)23 b Fm(!)15 b Fr(\012)j(is)g(an)f(extension)h(of)g Fn(g)r Fr(.)f(If)h Fn(x)f Fm(2)f Fr(\()p Fn(K)g Fm(\\)c Fn(T)7 b Fr(\))13 b Fm(\000)e Fn(R)p Fr(,)18 b(then)g Fn(x)1303 1616 y Fj(\000)p Fl(1)1373 1634 y Fm(2)e Ff(m)p Fr(,)i(so)g Fn(g)r Fr(\()p Fn(x)1628 1616 y Fj(\000)p Fl(1)1681 1634 y Fr(\))f(=)f(0,)h(hence)0 1703 y Fn(h)p Fr(\()p Fn(x)76 1685 y Fj(\000)p Fl(1)130 1703 y Fr(\))d(=)g(0)h(and)h Fn(h)p Fr(\()p Fn(x)p Fr(\))g(has)f(trouble)g(de\014ning)f(itself.)i(W) l(e)g(conclude)f(that)h(suc)o(h)f(troublesome)f Fn(x)p Fr('s)h(in)0 1773 y(\()p Fn(K)f Fm(\\)9 b Fn(T)e Fr(\))j Fm(\000)f Fn(R)16 b Fr(cannot)f(actually)h(exist,)g(so)f(w)o(e)g(m)o (ust)g(ha)o(v)o(e)g Fn(T)h Fm(\\)10 b Fn(K)17 b Fr(=)d Fn(R)p Fr(.)i(Th)o(us)e(the)i(second)f(theorem)0 1843 y(implies)i(the)i(\014rst,)e(and)h(w)o(e)g(ha)o(v)o(e)g(reduced)g(the)g (problem)f(to)i(establishing)e(Theorem)g(2.)h(T)l(o)g(do)h(so,)0 1913 y(w)o(e)d(\014rst)g(need)g(a)h(lemma:)100 2022 y Fp(Lemma)p Fr(:)g(Let)i Fn(B)i Fr(b)q(e)f(a)f(lo)q(cal)g(subring)e(of)i (a)g(\014eld)f(K,)h(with)f(maximal)g(ideal)h Ff(m)f Fm(\032)f Fn(B)r Fr(,)i(and)g(let)0 2092 y Fn(x)14 b Fm(2)g Fn(K)135 2074 y Fj(\003)158 2092 y Fr(.)j(Then)f Ff(m)p Fr([)p Fn(x)p Fr(])e Fm(6)p Fr(=)f Fn(B)r Fr([)p Fn(x)p Fr(])k(or)f Ff(m)p Fr([)p Fn(x)733 2074 y Fj(\000)p Fl(1)787 2092 y Fr(])e Fm(6)p Fr(=)g Fn(B)r Fr([)p Fn(x)950 2074 y Fj(\000)p Fl(1)1004 2092 y Fr(].)100 2201 y Fp(Pro)r(of)p Fr(:)g(Supp)q(ose)i(the)g(lemma)g(is)g(false.)g(Then)594 2304 y(1)e(=)g Fn(u)715 2311 y Fl(0)748 2304 y Fr(+)c Fn(u)826 2311 y Fl(1)848 2304 y Fn(x)i Fr(+)f Fm(\001)d(\001)g(\001)j Fr(+)g Fn(u)1086 2311 y Fk(m)1123 2304 y Fn(x)1151 2283 y Fk(m)1189 2304 y Fn(;)d(u)1240 2311 y Fk(i)1270 2304 y Fm(2)14 b Ff(m)565 2406 y Fr(1)f(=)h Fn(v)680 2413 y Fl(0)713 2406 y Fr(+)d Fn(v)787 2413 y Fl(1)810 2406 y Fn(x)838 2385 y Fj(\000)p Fl(1)903 2406 y Fr(+)g Fm(\001)d(\001)g (\001)j Fr(+)g Fn(v)1096 2413 y Fk(n)1123 2406 y Fn(x)1151 2385 y Fj(\000)p Fk(n)1210 2406 y Fn(;)d(v)1256 2413 y Fk(i)1286 2406 y Fm(2)14 b Ff(m)p Fn(:)0 2498 y Fr(Assume)i(that)h (the)g(ab)q(o)o(v)o(e)g Fn(m;)8 b(n)17 b Fr(are)f(minimal.)f(Also,)i (WLOG,)g(assume)e(that)i Fn(n)e Fm(\024)f Fn(m)p Fr(.)i(Multiplying)0 2568 y(the)h(second)f(equation)g(b)o(y)g Fn(x)547 2550 y Fk(n)575 2568 y Fr(,)g(w)o(e)h(ha)o(v)o(e)522 2670
y Fn(v)546 2677 y Fl(1)569 2670 y Fn(x)597 2649 y Fk(n)p Fj(\000)p Fl(1)687 2670 y Fr(+)10 b Fn(v)760 2677 y Fl(2)783 2670 y Fn(x)811 2649 y Fk(n)p Fj(\000)p Fl(2)901 2670 y Fr(+)g Fm(\001)e(\001)g(\001)k Fr(+)e Fn(v)1093 2677 y Fk(n)1134 2670 y Fr(=)k(\(1)d Fm(\000)g Fn(v)1316 2677 y Fl(0)1339 2670 y Fr(\))p Fn(x)1386 2649 y Fk(n)1414 2670 y Fn(;)950 2790 y Fr(10)p eop %%Page: 11 11 11 10 bop 0 50 a Fr(where)13 b(1)5 b Fm(\000)g Fn(v)239 57 y Fl(0)261 50 y Fr(,)14 b(not)f(living)g(in)g(the)h(maximal)f (ideal,)f(is)h(forced)h(to)g(b)q(e)f(in)g Fn(B)1386 32 y Fj(\003)1410 50 y Fr(.)g(Therefore)g(w)o(e)g(can)h(divide)0 120 y(the)j(equation)f(through)g(b)o(y)g(\(1)11 b Fm(\000)g Fn(v)676 127 y Fl(0)698 120 y Fr(\))17 b(to)g(get)g(a)g(new)f (equation:)608 239 y Fn(v)634 219 y Fj(0)632 252 y Fl(1)654 239 y Fn(x)682 219 y Fk(n)p Fj(\000)p Fl(1)772 239 y Fr(+)11 b Fn(v)848 219 y Fj(0)846 252 y Fl(2)868 239 y Fn(x)896 219 y Fk(n)p Fj(\000)p Fl(2)986 239 y Fr(+)g Fm(\001)d(\001)g(\001)j Fr(+)g Fn(v)1181 219 y Fj(0)1179 252 y Fk(n)1220 239 y Fr(=)j Fn(x)1301 219 y Fk(n)1328 239 y Fn(:)0 360 y Fr(Multiplying)h(this)h(through)g(b)o(y)g Fn(x)654 342 y Fk(m)p Fj(\000)p Fk(n)765 360 y Fr(giv)o(es)g(the)g (relation)643 480 y Fn(v)669 459 y Fj(0)667 492 y Fl(1)690 480 y Fn(x)718 459 y Fk(m)p Fj(\000)p Fl(1)818 480 y Fr(+)11 b Fm(\001)d(\001)g(\001)j Fr(+)g Fn(v)1013 459 y Fj(0)1011 492 y Fk(n)1038 480 y Fn(x)1066 459 y Fk(m)p Fj(\000)p Fk(n)1174 480 y Fr(=)j Fn(x)1255 459 y Fk(m)1293 480 y Fn(:)0 600 y Fr(No)o(w)j(if)g(w)o(e)g(plug)g(the)g (left-hand-side)f(of)h(this)g(expression)e(in)o(to)i(our)g(putativ)o (ely)g(minimal)e(equation)0 670 y(in)o(v)o(olving)h Fn(x)242 652 y Fk(m)281 670 y Fr(,)h(w)o(e)h(get)g(a)g(new)g(equation)f(whic)o (h)g(con)o(tradicts)g(the)h(minimalit)o(y)e(of)i Fn(m)p Fr(.)g(This)f(pro)o(v)o(es)0 740 y(the)g(lemma.)100 849 y Fp(Pro)r(of)g(of)h(Theorem)f(2)p Fr(\(See)g(also)f(A)o(tiy)o (ah-MacDonald,)e(p.65\):)100 959 y(Since)21 b Fn(g)r Fr(\()p Fn(B)r Fr(\))h(is)f(a)g(subring)e(of)j(a)f(\014eld)g(and)g (therefore)g(an)g(in)o(tegral)f(domain,)g(the)i(ideal)f Ff(m)g Fr(=)0 1028 y(k)o(er\()p Fn(g)r Fr(\))f(is)g(prime.)e(No)o(w)i (if)h Fn(s)f Fm(2)g Fn(B)c Fm(\000)d Ff(m)p Fr(,)20 b(then)g Fn(g)r Fr(\()p Fn(s)p Fr(\))h(will)f(not)g(b)q(e)g(zero,)g(so)g(w)o(e)g (can)f(extend)i Fn(g)h Fr(to)e(a)0 1098 y(homomorphism)14 b Fn(g)379 1080 y Fj(0)407 1098 y Fr(:)h Fn(B)474 1105 y Fe(m)518 1098 y Fm(!)g Fr(\012)i(b)o(y)g(putting)g Fn(g)910 1080 y Fj(0)923 1098 y Fr(\()p Fn(b=s)p Fr(\))g(=)d Fn(g)r Fr(\()p Fn(b)p Fr(\))p Fn(=g)r Fr(\()p Fn(s)p Fr(\))19 b(for)e(all)g Fn(b)e Fm(2)g Fn(B)r(;)8 b(s)16 b Fm(2)f Fn(B)f Fm(\000)d Ff(m)p Fr(.)18 b(But)0 1168 y(the)f(pair)f(\()p Fn(B)r(;)8 b(g)r Fr(\))17 b(w)o(as)f(c)o(hosen)f (to)i(b)q(e)g(maximal,)f(so)g(w)o(e)g(m)o(ust)g(ha)o(v)o(e)g Fn(B)g Fr(=)e Fn(B)1452 1175 y Fe(m)1481 1168 y Fr(,)j(and)f(therefore) g Fn(B)j Fr(is)d(a)0 1238 y(lo)q(cal)i(ring)g(with)g Ff(m)g Fr(as)g(its)g(maximal)g(ideal.)f(It)i(remains)e(to)h(sho)o(w)f (that)i(if)f Fn(x)g Fm(6)p Fr(=)e(0)i(is)g(an)g(elemen)o(t)g(of)0 1307 y Fn(K)t Fr(,)e(then)f(either)h Fn(x)e Fm(2)g Fn(B)k Fr(or)d Fn(x)562 1289 y Fj(\000)p Fl(1)630 1307 y Fm(2)f Fn(B)r Fr(.)i(Since)f(w)o(e)h(no)o(w)f(kno)o(w)g(that)h Fn(B)i Fr(is)e(a)f(lo)q(cal)h(ring,)f(w)o(e)g(can)h(apply)0 1377 y(our)k(lemma,)f(and)h(w)o(e)h(see)g(that)g(w)o(e)f(ma)o(y)g(as)g (w)o(ell)h(assume)e(that)i(the)g(ideal)f Ff(m)p Fr([)p
Fn(x)p Fr(])h(is)f(not)h(equal)g(to)0 1447 y Fn(B)40 1429 y Fj(0)68 1447 y Fr(=)14 b Fn(B)r Fr([)p Fn(x)p Fr(].)i(So)f(then)h Ff(m)p Fr([)p Fn(x)p Fr(])h(is)e(con)o(tained)g(in) g(a)h(maximal)f(ideal)g Ff(m)1270 1429 y Fj(0)1301 1447 y Fr(of)h Fn(B)1397 1429 y Fj(0)1411 1447 y Fr(,)g(and)f(it)h(m)o(ust)f (b)q(e)h(the)g(case)0 1517 y(that)e Ff(m)143 1499 y Fj(0)162 1517 y Fm(\\)5 b Fn(B)16 b Fr(=)e Ff(m)p Fr(,)g(since)f(it)g(is)g(a)h (prop)q(er)e(ideal)h(of)h Fn(B)i Fr(and)d(con)o(tains)f Ff(m)p Fr(.)i(Therefore)e(the)i(em)o(b)q(edding)e(of)0 1586 y Fn(B)18 b Fr(in)c Fn(B)152 1568 y Fj(0)182 1586 y Fr(induces)g(an)h(em)o(b)q(edding)f(of)h(the)h(\014eld)e Fn(k)i Fr(=)d Fn(B)r(=)p Ff(m)j Fr(in)o(to)f(the)g(\014eld)g Fn(k)1455 1568 y Fj(0)1482 1586 y Fr(=)f Fn(B)1575 1568 y Fj(0)1589 1586 y Fn(=)p Ff(m)1652 1568 y Fj(0)1666 1586 y Fr(.)h(In)g(addition,)0 1656 y(w)o(e)k(ha)o(v)o(e)f Fn(k)221 1638 y Fj(0)252 1656 y Fr(=)g Fn(k)r Fr([)p 351 1629 29 2 v Fn(x)p Fr(],)g(where)p 572 1629 V 19 w Fn(x)h Fr(is)g(the)g(image)f(of)h Fn(x)h Fr(in)e Fn(k)1101 1638 y Fj(0)1115 1656 y Fr(,)h(hence)p 1288 1629 V 18 w Fn(x)h Fr(is)e(algebraic)g(o)o(v)o(er)g Fn(k)j Fr(and)d Fn(k)1884 1638 y Fj(0)1917 1656 y Fr(is)0 1726 y(then)h(a)g(\014nite)f (algebraic)g(extension)h(of)g Fn(k)r Fr(.)f(No)o(w)h(since)f Ff(m)h Fr(is)g(the)g(k)o(ernel)f(of)h Fn(g)r Fr(,)g(w)o(e)f(get)i(an)e (induced)0 1796 y(em)o(b)q(edding)p 251 1768 26 2 v 14 w Fn(g)d Fr(:)f Fn(k)h Fm(!)f Fr(\012.)h(Since)h(\012)f(is)g (algebraically)g(closed,)g(w)o(e)g(can)g(extend)p 1490 1768 V 16 w Fn(g)i Fr(to)f(an)g(em)o(b)q(edding)p 1910 1768 V 13 w Fn(g)1936 1776 y Fj(0)0 1865 y Fr(of)h Fn(k)85 1847 y Fj(0)114 1865 y Fr(in)o(to)f(\012.)h(If)f(w)o(e)g(no)o(w)g(comp) q(ose)p 709 1838 V 16 w Fn(g)734 1846 y Fj(0)765 1865 y Fr(with)g(the)h(natural)e(homomorphism)e(from)i Fn(B)1647 1847 y Fj(0)1678 1865 y Fr(to)i Fn(k)1767 1847 y Fj(0)1780 1865 y Fr(,)f(w)o(e)h(get)0 1935 y(a)f(new)g(map)g Fn(g)280 1917 y Fj(0)307 1935 y Fr(:)e Fn(B)375 1917 y Fj(0)403 1935 y Fm(!)f Fr(\012)k(whic)o(h)e(extends)h Fn(g)r Fr(.)g(Since)g(\()p Fn(B)r(;)8 b(g)r Fr(\))17 b(w)o(as)e(c)o(hosen)g(to)i(b)q(e)f(maximal,) f(w)o(e)h(m)o(ust)0 2005 y(ha)o(v)o(e)g Fn(B)155 1987 y Fj(0)183 2005 y Fr(=)e Fn(B)r Fr(,)i(hence)h Fn(x)d Fm(2)g Fn(B)19 b Fr(as)d(desired.)100 2114 y(One)g(application)f(of)i (these)f(results)g(is)g(in)g(the)h(follo)o(wing)e(useful)h(theorem.)100 2224 y Fp(Theorem)p Fr(:)h(Let)k Fn(A)f Fr(b)q(e)f(a)h(subring)e(of)h (a)h(\014eld)f Fn(K)t Fr(.)g(Then)g(the)h(in)o(tegral)f(closure)f(of)i Fn(A)g Fr(in)f Fn(K)24 b Fr(is)0 2294 y(equal)16 b(to)h(the)g(in)o (tersection)f(of)g(all)h(the)f(v)m(aluation)h(rings)e(of)h Fn(K)21 b Fr(con)o(taining)15 b Fn(A)p Fr(.)100 2403 y Fp(Pf)p Fr(:)21 b(\\)p Fm(\032)p Fr(":)g(Supp)q(ose)f Fn(x)i Fr(is)g(in)f(the)h(in)o(tegral)e(closure)h(of)g Fn(A)h Fr(in)f Fn(K)t Fr(.)h(Then)f(w)o(e)g(can)h(write)f Fn(x)1869 2385 y Fk(n)1911 2403 y Fr(+)0 2473 y Fn(a)26 2480 y Fk(n)p Fj(\000)p Fl(1)104 2473 y Fn(x)132 2455 y Fk(n)p Fj(\000)p Fl(1)223 2473 y Fr(+)11 b Fm(\001)d(\001)g(\001)k Fr(+)f Fn(a)419 2480 y Fl(0)457 2473 y Fr(=)k(0,)i(with)h Fn(a)708 2480 y Fk(i)740 2473 y Fm(2)d Fn(A)p Fr(.)j(W)l(e)g(w)o(an)o (t)e(to)i(kno)o(w)f(that)h Fn(x)d Fm(2)j Fr(an)o(y)f Fn(R)p Fr(.)g(No)o(w)h(the)f Fn(x)f Fr(=)f(0)0 2543 y(case)h(is)h (trivial.)f(Otherwise,)f(divide)h(the)h(ab)q(o)o(v)o(e)f(equation)g(b)o (y)h Fn(x)1243 2525 y Fk(n)p Fj(\000)p Fl(1)1338 2543 y Fr(to)g(get)512 2670 y Fn(x)d Fr(=)g Fm(\000)p Fn(a)672 2677 y Fk(n)p Fj(\000)p Fl(1)761 2670 y Fm(\000)d Fn(a)837
2677 y Fk(n)p Fj(\000)p Fl(2)915 2670 y Fn(x)943 2649 y Fj(\000)p Fl(1)1008 2670 y Fm(\000)g(\001)d(\001)g(\001)j(\000)g Fn(a)1203 2677 y Fl(0)1226 2670 y Fr(\()p Fn(x)1273 2649 y Fj(\000)p Fl(1)1327 2670 y Fr(\))1346 2639 y Fk(n)p Fj(\000)p Fl(1)1424 2670 y Fn(:)950 2790 y Fr(11)p eop %%Page: 12 12 12 11 bop 0 50 a Fr(Therefore)18 b Fn(x)g Fm(2)g Fn(A)p Fr([)p Fn(x)404 32 y Fj(\000)p Fl(1)458 50 y Fr(].)h(No)o(w)f(pic)o(k)h (an)o(y)f(v)m(aluation)h(ring)f Fn(R)p Fr(,)h(and)f(w)o(e)g(ha)o(v)o(e) g(t)o(w)o(o)h(cases.)f(If)h Fn(x)f Fm(2)g Fn(R)p Fr(,)0 120 y(then)h(w)o(e're)f(done.)g(If)h Fn(x)462 101 y Fj(\000)p Fl(1)534 120 y Fm(2)f Fn(R)p Fr(,)g(then)h Fn(x)f Fm(2)g Fn(A)p Fr([)p Fn(x)947 101 y Fj(\000)p Fl(1)1001 120 y Fr(])g Fm(\032)f Fn(R)p Fr(,)i(so)g(w)o(e're)f(also)g(done.)g(That)h (\014nishes)e(o\013)0 189 y(one)f(direction.)100 303 y(\\)p Fm(\033)p Fr(":)10 b(Supp)q(ose)g(w)o(e)h(ha)o(v)o(e)g(some)f Fn(x)i Fr(whic)o(h)e(is)h(NOT)h(in)o(tegral)e(o)o(v)o(er)g Fn(A)p Fr(.)i(W)l(e)f(w)o(ould)f(lik)o(e)h(to)h(construct)0 373 y(an)i Fn(R)g Fr(whic)o(h)f('misses')g Fn(x)p Fr(.)h(T)l(o)g(do)g (this,)g(w)o(e)g(lo)q(ok)g(at)h Fn(A)p Fr([)p Fn(x)1055 355 y Fj(\000)p Fl(1)1109 373 y Fr(].)f(No)o(w)g(it)g(cannot)g(b)q(e)h (that)f Fn(x)g Fm(2)h Fn(A)p Fr([)p Fn(x)1810 355 y Fj(\000)p Fl(1)1864 373 y Fr(],)f(or)0 443 y(else)h(m)o(ultiplying)e(b)o(y)h(an)h (appropriate)e(p)q(o)o(w)o(er)h(of)h Fn(x)h Fr(giv)o(es)e(an)h(in)o (tegralit)o(y)f(relation)g(for)g Fn(x)p Fr(.)i(Therefore)0 513 y Fn(x)28 495 y Fj(\000)p Fl(1)101 513 y Fr(is)i(not)h(in)g Fn(A)p Fr([)p Fn(x)384 495 y Fj(\000)p Fl(1)438 513 y Fr(])452 495 y Fj(\003)475 513 y Fr(,)g(so)f(there)h(is)g(a)g(maximal)e (ideal)i Ff(m)g Fr(of)g Fn(A)p Fr([)p Fn(x)1320 495 y Fj(\000)p Fl(1)1374 513 y Fr(])g(con)o(taining)f Fn(x)1680 495 y Fj(\000)p Fl(1)1734 513 y Fr(.)h(If)g(w)o(e)g(let)0 587 y(\012)j(=)p 118 544 211 2 v 21 w Fn(A)p Fr([)p Fn(x)197 573 y Fj(\000)p Fl(1)251 587 y Fr(])p Fn(=)p Ff(m)p Fr(,)f(then)h (there)f(is)g(a)g(ring)f(homomorphism)e Fn(f)27 b Fr(:)22 b Fn(A)p Fr([)p Fn(x)1345 569 y Fj(\000)p Fl(1)1399 587 y Fr(])g Fm(!)f Fr(\012,)g(and)g(w)o(e)g(can)g(then)0 657 y(\014nd)d(a)g(maximal)g(pair)g(\()p Fn(R;)8 b(g)r Fr(\))19 b(as)f(b)q(efore,)h(where)f Fn(R)h Fr(w)o(e)f(no)o(w)g(kno)o (w)h(to)g(b)q(e)g(a)f(v)m(aluation)h(ring)e(of)i Fn(K)0 727 y Fr(con)o(taining)d Fn(A)p Fr([)p Fn(x)322 709 y Fj(\000)p Fl(1)376 727 y Fr(],)i(and)f Fn(g)i Fr(is)f(a)f(homomorphism) d(from)j Fn(R)h Fr(to)g(\012)g(whic)o(h)f(extends)h Fn(f)5 b Fr(.)18 b(I)g(claim)f(that)0 797 y(this)d(is)g(the)h(desired)e Fn(R)h Fr(whic)o(h)g(cannot)g(con)o(tain)f Fn(x)p Fr(.)i(F)l(or)e Fn(g)r Fr(\()p Fn(x)1138 778 y Fj(\000)p Fl(1)1192 797 y Fr(\))h(=)g(0,)g(so)g(ha)o(ving)f Fn(x)i Fm(2)f Fn(R)g Fr(w)o(ould)f(cause)0 866 y(ma)s(jor)i(problems)g(if)i(one)f(tried)g (to)h(de\014ne)f Fn(g)r Fr(\()p Fn(x)p Fr(\).)h(This)f(pro)o(v)o(es)f (the)i(theorem.)100 980 y Fp(Note)p Fr(:)e(In)g(the)g(situations)f (that)i(w)o(e)f(will)g(b)q(e)g(concerned)g(with)g(in)g(this)g(course,)f (w)o(e)h(w)o(on't)f(really)0 1050 y(need)19 b(the)g(axiom)g(of)g(c)o (hoice,)g(since)f(w)o(e'll)h(b)q(e)g(giv)o(en)g(that)g Fn(K)k Fr(is)c(a)g(\014nite)g(extension)g(of)g(the)g(\014eld)g(of)0 1120 y(fractions)d(of)h Fn(A)p Fr(.)100 1234 y(W)l(e)g(conclude)g(with) g(a)g(v)m(ague)g(set)h(of)f(remarks.)f(Last)h(time,)g(w)o(e)g(found)f (a)h(bijectiv)o(e)h(corresp)q(on-)0 1304 y(dence:)100 1418 y(non)o(trivial)d(v)m(aluation)h(rings)f(of)i Fn(k)r Fr(\()p Fn(t)p Fr(\))g(o)o(v)o(er)e Fn(k)j Fm( )-8 b(!)16 b Fn(f)k Fm(2)14 b Fn(k)r Fr([)p Fn(t)p Fr(])i(monic,)f(irreducible)g
Fm([1)p Fr(.)100 1532 y(Recall)h(that)h(if)f Fn(R)e Fm($)g Fn(f)5 b Fr(,)17 b(then)g Fn(R=)p Ff(m)808 1519 y Fm(\030)808 1534 y Fr(=)860 1532 y Fn(k)r Fr([)p Fn(t)p Fr(])p Fn(=)p Fr(\()p Fn(f)5 b Fr(\).)17 b(If)g Fn(R)d Fm($)g(1)p Fr(,)i(then)h Fn(R=)p Ff(m)1532 1519 y Fm(\030)1532 1534 y Fr(=)1584 1532 y Fn(k)r Fr(.)100 1648 y(No)o(w)g(if)h Fn(k)f Fr(=)p 359 1608 28 2 v 16 w Fn(k)r Fr(,)g(then)h Fn(f)23 b Fr(irreducible)16 b Fm(,)g Fn(f)21 b Fr(=)16 b Fn(t)c Fm(\000)f Fn(\013;)d(\013)17 b Fm(2)f Fn(k)r Fr(.)h(Then)g(w)o(e)h(ha)o(v)o(e)f(v)m(alutation)g (rings)0 1718 y(of)g Fn(k)r Fr(\()p Fn(t)p Fr(\))g(o)o(v)o(er)e Fn(k)j Fm( )-8 b(!)16 b Fr(the)h(pro)s(jectiv)o(e)f(line)g(o)o(v)o(er)g Fn(k)r Fr(.)100 1832 y(F)l(or)i(general)i Fn(k)r Fr(,)f(w)o(e)h(ha)o(v) o(e)f(that)h(the)g(pro)s(jectiv)o(e)g(line)f(is)h(in)f(corresp)q (ondence)g(with)h(the)g(set)g(of)0 1902 y Fn(R)p Fr('s)d(with)f (residue)g(class)g(\014eld)g(=)h Fn(k)r Fr(.)f(F)l(or)g(example,)g(if)h Fn(k)f Fr(=)e Fp(R)p Fr(,)i(then)h(irreducibles)d(of)j(degree)g(1)g Fm( )-8 b(!)0 1972 y Fr(real)15 b(pro)s(jectiv)o(e)g(line.)g(F)l(or)f (irreducibles)f(of)j(degree)f(2,)g(w)o(e)g(need)g(to)h(lo)q(ok)f(at)h (the)g(orbit)f(of)g(Gal\()p Fp(C)p Fn(=)p Fp(R)p Fr(\))0 2041 y(acting)21 b(on)g(the)h(complex)e(pro)s(jectiv)o(e)h(line.)g(A)h (general)e(strategy)i(in)e(this)h(course)g(will)g(b)q(e)g(that)h(to)0 2111 y(understand)13 b(the)j(p)q(oin)o(ts)e(on)h(some)g(curv)o(e)f Fn(C)19 b Fr(whic)o(h)14 b(maps)g(to)h(the)h(pro)s(jectiv)o(e)f(line,)f (w)o(e)h(will)g(w)o(an)o(t)g(to)0 2181 y(understand)g(the)h(set)h(of)g (v)m(aluation)f(rings)f(extending)i Fn(k)r Fr(\()p Fn(t)p Fr(\).)100 2295 y(Next)g(time:)g Fn(e)p Fr('s)f(and)g Fn(f)5 b Fr('s.)100 2369 y Fp(Lecture)20 b(6)c Fr(\(Septem)o(b)q(er)g (11,)g(1995\).)0 2444 y(Scrib)q(e:)g(Diane)g(Maclagan)100 2600 y(In)k(this)g(lecture)g(w)o(e)g(study)g(a)h(\014eld)f Fn(K)t Fr(,)g(a)g(v)m(aluation)g(ring)g Fn(R)g Fm(\022)g Fn(K)t Fr(,)h(and)e(an)h(extension)h(\014eld)0 2670 y Fn(L)14 b Fm(\023)f Fn(K)t Fr(.)k(W)l(e)f(are)h(in)o(terested)e(in)h(v) m(aluation)h(rings)e Fn(T)21 b Fm(\022)13 b Fn(L)k Fr(suc)o(h)e(that)i Fn(T)h Fm(\\)11 b Fn(K)18 b Fr(=)c Fn(R)p Fr(.)950 2790 y(12)p eop %%Page: 13 13 13 12 bop 100 50 a Fp(De\014nition:)21 b Fr(The)c(residue)e(class)i (degree,)f Fn(f)5 b Fr(\()p Fn(T)h(=R)p Fr(\),)18 b(is)f(de\014ned)f (to)i(b)q(e)f Fn(f)j Fr(=)15 b([)p Fn(T)6 b(=m)1709 57 y Fk(T)1754 50 y Fr(:)15 b Fn(R=m)1890 57 y Fk(R)1922 50 y Fr(].)0 120 y(W)l(e)j(are)f(assuming)e(here)i(that)g Fn(m)649 127 y Fk(R)697 120 y Fr(=)e Fn(m)795 127 y Fk(T)838 120 y Fm(\\)c Fn(K)t Fr(.)18 b(This)e(is)h(a)g(consequence)g(of)h Fn(T)g Fm(\\)12 b Fn(K)19 b Fr(=)c Fn(R)p Fr(.)j Fn(f)5 b Fr(\()p Fn(T)h(=R)p Fr(\))0 189 y(is)16 b(sometimes)f(also)h(written) h Fn(f)5 b Fr(\()p Fn(L=K)t Fr(\).)17 b(W)l(e)g(are)f(in)o(terested)g (in)g(the)h(case)f(where)g Fn(f)23 b Fr(is)16 b(\014nite.)100 299 y Fp(De\014nition:)28 b Fr(The)22 b(rami\014cation)g(index,)g Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\),)24 b(is)f(de\014ned)f(to)h(b)q(e)h Fn(e)h Fr(=)f([\001)g(:)h(\000],)e(where)0 369 y(\000)14 b(=)g Fn(K)144 351 y Fj(\003)166 369 y Fn(=R)229 351 y Fj(\003)269 369 y Fr(and)i(\001)d(=)h Fn(L)508 351 y Fj(\003)531 369 y Fn(=T)592 351 y Fj(\003)614 369 y Fr(.)j([\001)c(:)h(\000])f(=)h Fm(j)p Fn(L)901 351 y Fj(\003)924 369 y Fn(=K)995 351 y Fj(\003)1017 369 y Fn(:T)1067 351 y Fj(\003)1090 369 y Fm(j)p Fr(.)100 478 y(The)f(geometric)g(in)o(terpretation)g(of)h(this)f(is)g(as)h(follo)o (ws:)e(If)i Fn(C)j Fr(is)c(a)h(curv)o(e)f(with)h(function)f(\014eld)g Fn(K)t Fr(,)0 548 y(and)k Fn(L)p Fr(,)g(a)h(\014nite)f(extension)g(of)h
Fn(K)t Fr(,)f(is)g(the)h(function)f(\014eld)g(of)g(some)g(curv)o(e)g Fn(D)q Fr(,)h(then)f Fn(e)h Fr(is)f(a)g(measure)0 618 y(of)j(n)o(um)o(b)q(er)f(of)h(p)q(oin)o(ts)f(on)h Fn(D)i Fr(ab)q(o)o(v)o(e)e Fn(C)t Fr(.)f(If)i Fn(e)f Fr(=)g(1)g(w)o(e)g(sa)o (y)g(that)h(the)f(p)q(oin)o(t)g(is)g(unrami\014ed,)e(while)0 687 y(if)j Fn(e)g(>)g Fr(1,)g(it)g(is)g(rami\014ed.)e(W)l(e)i(usually)f (exp)q(ect)i(only)e(\014nitely)h(man)o(y)f(p)q(oin)o(ts)g(where)h (rami\014cation)0 757 y(o)q(ccurs.)100 827 y(Note)13 b(that)f(for)g(algebraically)f(closed)h(base)g(\014elds,)f Fn(f)19 b Fr(=)14 b(1.)e(If)g(the)h(base)f(\014eld)f(is)h(not)g (algebraically)0 897 y(closed,)18 b(w)o(e)h(can)g(think)g(of)h(the)f(v) m(aluation)g(rings)f(not)h(as)g(p)q(oin)o(ts)f(on)h(the)g(curv)o(e,)f (but)h(as)g(conjugacy)0 966 y(classes)d(of)g(p)q(oin)o(ts,)g(so)g(they) h(ha)o(v)o(e)f(size,)g(and)g(if)h Fn(Q)g Fr(is)f(a)g(p)q(oin)o(t)g(o)o (v)o(er)g Fn(P)7 b Fr(,)16 b(then)h Fn(f)i Fr(=)14 b Fm(j)p Fn(Q)p Fm(j)p Fn(=)p Fm(j)p Fn(P)7 b Fm(j)p Fr(.)100 1036 y(W)l(e)16 b(will)h(sho)o(w)e(that)i(for)f(curv)o(es)753 1104 y Fh(X)631 1220 y Fk(Q)h Fr(o)o(v)o(er)e(giv)o(en)i Fk(P)955 1151 y Fn(ef)i Fr(=)14 b Fn(n)g Fr(=)f([)p Fn(L)h Fr(:)f Fn(K)t Fr(])0 1328 y(This)j(is)g(not)g(true)h(of)f(general)g(v)m (aluation)g(rings.)f(W)l(e)i(start)g(with)f(the)h(follo)o(wing)e(prop)q (osition:)100 1437 y Fp(Prop)r(osition:)35 b Fr(If)17 b Fn(L)f Fr(is)g(\014nite)h(o)o(v)o(er)e Fn(K)21 b Fr(then)16 b(the)h(pro)q(duct)f Fn(ef)k Fr(=)13 b Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))p Fn(f)f Fr(\()p Fn(T)h(=R)p Fr(\))16 b Fm(\024)e Fr([)p Fn(L)f Fr(:)h Fn(K)t Fr(])100 1547 y Fp(Pro)r(of:)46 b Fr(Let)24 b Fn(B)i Fm(\022)e Fn(T)30 b Fr(map)21 b(bijectiv)o(ely)j(to)e(a)h(basis)f(of)g Fn(T)6 b(=m)1355 1554 y Fk(T)1409 1547 y Fr(o)o(v)o(er)22 b Fn(R=m)1629 1554 y Fk(R)1661 1547 y Fr(,)h(noting)f(that)0 1617 y Fn(B)k Fm(\022)d Fn(T)162 1599 y Fj(\003)185 1617 y Fr(.)f(Let)h Fn(S)i Fm(\022)e Fn(L)468 1599 y Fj(\003)513 1617 y Fr(b)q(e)g(a)f(set)g(of)h(coset)f(represen)o(tativ)o(es)f(mo)q (d)g Fn(K)1419 1599 y Fj(\003)1442 1617 y Fn(:T)1492 1599 y Fj(\003)1515 1617 y Fr(.)h(So)g(the)g(cardinalit)o(y)0 1686 y(of)e Fn(B)i Fr(is)d Fn(f)5 b Fr(,)20 b(and)f(the)g(cardinalit)o (y)f(of)i Fn(S)i Fr(is)d Fn(e)p Fr(.)h(It)g(su\016ces)e(to)i(pro)o(v)o (e)e(that)i(\()p Fn(b:s)p Fr(\))1536 1695 y Fk(b)p Fj(2)p Fk(B)r(;s)p Fj(2)p Fk(S)r Fl(\))1735 1686 y Fr(is)f(linearly)0 1756 y(indep)q(enden)o(t)c(o)o(v)o(er)h Fn(K)t Fr(.)100 1824 y(Supp)q(ose)297 1787 y Fh(P)349 1799 y Fk(<)p Fj(1)349 1839 y Fk(b)p Fj(2)p Fk(B)r(;s)p Fj(2)p Fk(S)r Fl(\))536 1824 y Fn(k)562 1831 y Fk(bs)601 1824 y Fn(bs)h Fr(=)g(0)h(where)g(not) g(all)h(the)f Fn(k)1182 1831 y Fk(bs)1238 1824 y Fr(=)e(0.)j(Cho)q(ose) f Fn(b)1547 1831 y Fl(0)1569 1824 y Fn(s)1592 1831 y Fl(0)1634 1824 y Fr(with)g Fn(k)1775 1831 y Fk(b)1793 1836 y Fd(0)1812 1831 y Fk(s)1831 1836 y Fd(0)1869 1824 y Fm(6)p Fr(=)f(0)0 1899 y(and)f Fn(v)r Fr(\()p Fn(k)168 1906 y Fk(b)186 1911 y Fd(0)205 1906 y Fk(s)224 1911 y Fd(0)262 1899 y Fr(minimal.)f(\()p Fn(v)h Fr(:)e Fn(L)591 1881 y Fj(\003)627 1899 y Fm(!)g Fr(\001,)i(and)g(w)o(e)g(lea)o(v)o(e)g (out)h Fn(b)g Fr(b)q(ecause)f(it)h(is)f(a)h(unit\).)648 1975 y Fh(X)646 2082 y Fk(b)p Fj(2)p Fk(B)730 2022 y Fn(k)756 2029 y Fk(bs)793 2034 y Fd(0)814 2022 y Fn(bs)858 2029 y Fl(0)892 2022 y Fr(+)983 1975 y Fh(X)986 2075 y Fc(b)p Fi(2)p Fc(B)948 2103 y(s)p Fi(2)p Fc(S)q(s)p Fi(6)p Fd(=)p Fc(s)1071 2110 y Fd(0)1104 2022 y Fn(k)1130 2029 y Fk(bs)1168 2022 y Fn(bs)e Fr(=)f(0)0 2209 y(No)o(w)i(divide)h(b) o(y)f Fn(k)357 2216 y Fk(b)375 2221 y Fd(0)394 2216 y
Fk(s)413 2221 y Fd(0)434 2209 y Fn(s)457 2216 y Fl(0)480 2209 y Fr(.)616 2313 y Fh(X)614 2420 y Fk(b)p Fj(2)p Fk(B)704 2327 y Fn(k)730 2334 y Fk(bs)767 2339 y Fd(0)788 2327 y Fn(bs)832 2334 y Fl(0)p 704 2349 152 2 v 705 2394 a Fn(k)731 2401 y Fk(b)749 2406 y Fd(0)768 2401 y Fk(s)787 2406 y Fd(0)809 2394 y Fn(s)832 2401 y Fl(0)872 2360 y Fr(+)963 2313 y Fh(X)966 2413 y Fc(b)p Fi(2)p Fc(B)928 2442 y(s)p Fi(2)p Fc(S)q(s)p Fi(6)p Fd(=)p Fc(s)1051 2449 y Fd(0)1110 2327 y Fn(k)1136 2334 y Fk(bs)1174 2327 y Fn(bs)p 1090 2349 150 2 v 1090 2394 a(k)1116 2401 y Fk(b)1134 2406 y Fd(0)1153 2401 y Fk(s)1172 2406 y Fd(0)1193 2394 y Fn(s)1216 2401 y Fl(0)1259 2360 y Fr(=)d(0)100 2531 y(All)h(the)h(terms)f(ha)o(v)o(e)f(p)q(ositiv)o(e)i(v)m(aluation,) f(so)g(b)q(elong)g(to)g Fn(T)7 b Fr(.)15 b(All)f(the)h(terms)f(in)g (the)g(second)g(sum)0 2600 y(ha)o(v)o(e)i(non-zero)g(v)m(aluation)h(b)q (ecause)f Fn(s)i Fr(and)e Fn(s)881 2607 y Fl(0)921 2600 y Fr(are)g(in)h(di\013eren)o(t)f(cosets)h(mo)q(d)g Fn(T)1554 2582 y Fj(\003)1576 2600 y Fn(K)1622 2582 y Fj(\003)1645 2600 y Fr(,)g(and)f(th)o(us)g(are)0 2670 y(in)g Fp(m)107 2677 y Fk(T)137 2670 y Fr(.)950 2790 y(13)p eop %%Page: 14 14 14 13 bop 100 52 a Fr(T)l(aking)16 b(this)g(sum)f(mo)q(d)h Fp(m)629 59 y Fk(T)676 52 y Fr(the)h(second)f(sum)f(disapp)q(ears,)g (so,)h(writing)f Fn(\025)1563 59 y Fk(b)1597 52 y Fr(=)1665 24 y Fk(k)1686 29 y Fc(bs)1719 36 y Fd(0)1741 24 y Fk(s)1760 29 y Fd(0)p 1656 41 134 2 v 1656 70 a Fk(k)1677 75 y Fc(b)1693 82 y Fd(0)1712 75 y Fc(s)1729 82 y Fd(0)1751 70 y Fk(s)1770 75 y Fd(0)1795 52 y Fr(,)731 157 y Fh(X)729 264 y Fk(b)p Fj(2)p Fk(B)813 204 y Fn(\025)842 211 y Fk(b)862 204 y Fr(\()p Fn(b)i Fr(mo)q(d)g Fp(m)1081 211 y Fk(T)1111 204 y Fr(\))d(=)f(0)100 376 y(This)i(means)h(that)h Fn(\025)503 383 y Fk(b)537 376 y Fr(=)d(0)8 b Fm(8)p Fn(\025)681 383 y Fk(b)716 376 y Fr(since)17 b(the)f Fn(b)h Fr(are)g(linearly)e(indep)q(enden)o(t,)h(but)g Fn(\025)1641 383 y Fk(b)1659 388 y Fd(0)1695 376 y Fr(=)d(1.)k(This)f (is)0 446 y(a)h(con)o(tradiction,)e(so)h(the)g(result)g(follo)o(ws.)100 556 y Fp(Corollary:)39 b Fr(If)19 b Fn(K)j Fr(is)c(a)g(\014nitely)g (generated)g(\014eld)f(extension)h(of)g(a)h(\014eld)e Fn(k)j Fr(of)e(transcendence)0 625 y(degree)h(1)g(\(trdeg)335 638 y Fk(k)359 625 y Fn(K)j Fr(=)c(1\))i(then)f(ev)o(ery)g(non)f (trivial)h(v)m(aluation)g(ring)f(of)h Fn(K)q(=k)i Fr(is)e(discrete)f (and)h(the)0 695 y(in)o(tersection)14 b(of)h(all)g(these)g(v)m (aluation)g(rings)f(equals)h Fm(f)p Fn(\013)f Fm(2)g Fn(K)j Fr(:)d Fn(\013)h Fr(is)g(a)g(algebraic)f(o)o(v)o(er)h Fn(k)r Fm(g)f Fr(and)h(this)g(is)0 765 y(a)i(\014nite)f(extension)g(of) h Fn(k)r Fr(.)100 875 y Fp(Pro)r(of:)26 b Fr(Let)14 b(T)f(b)q(e)g(a)g (non)o(trivial)e(v)m(aluation)i(ring)f(of)h Fn(K)q(=k)r Fr(,)g(and)f(let)h(R)g(b)q(e)g(its)g(restriction)f(to)i Fn(k)r Fr(\()p Fn(t)p Fr(\))0 944 y(\()p Fn(R)i Fr(=)g Fn(T)j Fm(\\)12 b Fn(k)r Fr(\()p Fn(t)p Fr(\)\).)18 b(Then)g Fn(R)g Fr(is)f(non)o(triv)m(al)g(b)o(y)g(exercise)h(4)f(of)h(the)g (homew)o(ork,)e(and)i(so)f(R)g(is)h(discrete,)0 1014 y(as)e(w)o(e)g(sho)o(w)o(ed)f(earlier)h(that)h(ev)o(ery)f(v)m(aluation) h(ring)e(of)i Fn(k)r Fr(\()p Fn(t)p Fr(\))g(w)o(as)e(discrete.)100 1084 y(Let)g(\001)f(b)q(e)g(the)h(v)m(alue)g(group)e(of)i Fn(K)j Fr(with)c(resp)q(ect)h(to)g Fn(T)7 b Fr(.)14 b(\001)g(is)g(a)g (totally)h(ordered)e(ab)q(elian)h(group)0 1153 y(con)o(taining)h Fp(Z)p Fr(,)i(and)e(is)i(of)f(\014nite)h(index)f Fn(e)h Fr(there,)f(so)g(\001)d(=)h Fp(Z)p Fr(.)i Fn(T)24 b Fr(is)16
b(th)o(us)g(discrete.)100 1223 y(No)o(w)376 1246 y Fh(\\)439 1293 y Fn(T)21 b Fr(=)674 1246 y Fh(\\)660 1346 y Fc(T)t Fi(\033)p Fc(k;)548 1391 y(T)t Fr(v)m(al.)c(ring)e(of)i Fc(K)869 1293 y Fn(T)k Fr(=)14 b Fm(f)p Fr(in)o(tegral)h(closure)g(of)i Fn(k)h Fr(in)e Fn(K)t Fm(g)0 1488 y Fr(Since)g Fn(k)i Fr(is)e(a)h(\014eld,)f(this)g(is)g(the)h(algebraic)e(elemen)o(ts.)100 1558 y(W)l(e)j(no)o(w)e(need)i(to)g(sho)o(w)e(that)i([)p Fm(\\)p Fn(T)23 b Fr(:)16 b Fn(k)r Fr(])f Fn(<)g Fm(1)p Fr(.)j(T)l(ak)o(e)f(an)o(y)g Fn(T)7 b Fr(.)18 b Fn(T)6 b(=)p Fp(m)1419 1565 y Fk(T)1466 1558 y Fr(is)17 b(of)h(\014nite)g (degree)f(o)o(v)o(er)0 1627 y Fn(R=)p Fp(m)112 1634 y Fk(R)144 1627 y Fr(,)k(whic)o(h)f(w)o(e)h(kno)o(w)g(from)g(last)g(time) g(is)g(o)o(v)o(er)g(\014nite)g(degree)g(o)o(v)o(er)f Fn(k)r Fr(,)h(so)g Fn(T)6 b(=)p Fp(m)1687 1634 y Fk(T)1738 1627 y Fr(is)21 b(a)g(\014nite)0 1697 y(extension)c(of)g Fn(k)r Fr(.)g(W)l(riting)f Fn(l)f Fr(=)604 1660 y Fh(T)654 1697 y Fn(T)7 b Fr(,)17 b Fn(l)e Fm(\022)g Fn(T)7 b Fr(,)17 b(and)f(it)h(injects)g(in)o(to)g Fn(T)6 b(=)p Fp(m)1387 1704 y Fk(T)1417 1697 y Fr(,)17 b(so)f(w)o(e)h(ha)o(v)o(e)f(em)o(b)q (edded)g Fn(l)0 1769 y Fr(in)g(a)h(\014nite)f(extension)g(of)h Fn(k)r Fr(,)f(so)g([)p Fn(l)f Fr(:)e Fn(k)r Fr(])g Fn(<)h Fm(1)p Fr(,)i(where)g Fn(l)f Fr(=)p 1109 1728 28 2 v 14 w Fn(k)d Fm(\\)f Fn(K)t Fr(.)100 1838 y Fn(l)21 b Fr(is)g(the)g Fq(exact)h(\014eld)g(of)g(c)m(onstants)p Fr(.)g(This)e(phrase)f(is)i(usually)f(used)g(when)g(w)o(e)h(sa)o(y)f (\\)p Fn(k)i Fr(is)f(the)0 1908 y(exact)c(\014eld)f(of)h(constan)o(ts") f(meaning)f Fn(k)g Fr(=)f Fn(l)q Fr(.)100 1978 y(Pro)s(jectiv)o(e)i (curv)o(es)g(cannot)g(b)q(e)h(adequately)g(de\014ned)f(b)o(y)g(rings,)f (as)h(ev)o(ery)h(non)f(constan)o(t)g(func-)0 2048 y(tion)g(has)g(p)q (oles)g(and)g(zero)q(es.)100 2117 y(Let)24 b Fn(R)g Fr(b)q(e)f(a)h (discrete)f(v)m(aluation)g(ring)f(in)h(a)h(\014eld)f Fn(K)k Fr(with)c(maximal)g(ideal)g Fp(m)p Fr(.)f(Then)h(the)0 2187 y(completion)16 b(of)g Fn(R)h Fr(is)226 2336 y(^)216 2349 y Fn(R)d Fr(=)19 b(lim)320 2379 y Fm( )-8 b(\000)409 2349 y Fn(R=)p Fp(m)521 2328 y Fk(n)561 2349 y Fr(=)13 b Fm(ff)p Fn(a)689 2356 y Fk(n)716 2349 y Fm(g)h(2)820 2287 y Fj(1)808 2301 y Fh(Y)802 2407 y Fk(n)p Fl(=0)886 2349 y Fn(R=)p Fp(m)998 2328 y Fk(n)1037 2349 y Fr(:)g Fm(8)p Fn(n)f Fm(\025)g Fr(0)k Fn(a)1257 2356 y Fk(n)1301 2349 y Fr(mo)q(d)f Fp(m)1462 2328 y Fk(n)p Fj(\000)p Fl(1)1552 2349 y Fr(=)e Fn(a)1631 2356 y Fk(n)p Fj(\000)p Fl(1)1709 2349 y Fm(g)0 2529 y Fr(This)h(is)h(a)g(pro)s(jectiv)o(e)f (limit.)578 2516 y(^)567 2529 y Fn(R)h Fr(is)g(a)g(ring,)f(and)g(there) h(is)f(a)h(natural)f(injectiv)o(e)i(ring)e(homomorphis)o(m)0 2600 y Fn(R)f Fm(!)126 2588 y Fr(^)116 2600 y Fn(R)p Fr(.)100 2670 y(Recommended)g(reading)i(for)g(this)g(section)g(is)g (the)h(relev)m(an)o(t)g(c)o(hapters)e(of)i(A)o(tiy)o(ah-Macdonald.)950 2790 y(14)p eop %%Page: 15 15 15 14 bop 100 50 a Fp(Example:)37 b Fr(Let)18 b Fn(k)i Fr(b)q(e)e(a)f(\014eld,)g Fn(R)f Fr(=)g Fn(k)r Fr([)p Fn(t)p Fr(])924 59 y Fl(\()p Fk(t)p Fl(\))971 50 y Fr(.)i(Then)f Fn(R=)p Fp(m)1246 32 y Fk(n)1288 50 y Fr(=)e Fn(k)r Fr([)p Fn(t)p Fr(])p Fn(=)p Fr(\()p Fn(t)1478 32 y Fk(n)1505 50 y Fr(\))j(and)1651 37 y(^)1640 50 y Fn(R)e Fr(=)f Fn(k)r Fr([[)p Fn(t)p Fr(]],)i(the)0 120 y(p)q(o)o(w)o(er)e(series)h (ring.)100 190 y Fp(Prop)r(erties)i(of)456 177 y Fr(^)446 190 y Fn(R)11 248 y Fr(^)0 260 y Fn(R)d Fr(is)f(a)g(domain.)f(It)i(is)f (a)h(discrete)f(v)m(aluation)g(ring)g(in)g(its)g(\014eld)g(of)h
(fractions,)f(whic)o(h)f(is)1667 248 y(^)1654 260 y Fn(K)18 b Fr(=)c Fn(K)c Fm(\012)1858 267 y Fk(R)1909 248 y Fr(^)1898 260 y Fn(R)p Fr(.)0 344 y Fn(e)p Fr(\()53 331 y(^)42 344 y Fn(R)q(=R)p Fr(\))24 b(=)g Fn(f)5 b Fr(\()310 331 y(^)298 344 y Fn(R)q(=R)p Fr(\))25 b(=)e(1.)574 318 y(^)579 331 y(^)568 344 y Fn(R)h Fr(=)703 331 y(^)693 344 y Fn(R)p Fr(,)e(and)g Fn(R)h Fr(is)f(complete)h(if)f(the)h(natural)f(map)f Fn(R)k Fm(!)1791 331 y Fr(^)1780 344 y Fn(R)e Fr(is)f(an)0 414 y(isomorphism.)0 538 y Fp(Lecture)e(7)c Fr(\(Septem)o(b)q(er)g(13,) h(1995\).)0 608 y(Scrib)q(e:)f(W)l(a)o(yne)g(Whitney)100 732 y(Recall)11 b(that)i(last)e(time)h(w)o(e)g(w)o(ere)f(discussing)f (extensions)i(of)g(v)m(aluation)f(rings:)g(giv)o(en)g(a)h(v)m(aluation) 0 802 y Fn(R)17 b Fr(with)f(\014eld)g(of)h(fractions)f Fn(K)t Fr(,)g(and)g(an)g(extension)g(of)h(\014elds)f Fn(L)d Fm(\033)h Fn(K)t Fr(,)i(w)o(e)h(de\014ned)e(a)i(v)m(aluation)f (ring)0 871 y Fn(T)32 b Fm(\032)25 b Fn(L)e Fr(to)g(b)q(e)h(an)f (extension)g(of)g Fn(R)h Fr(if)f Fn(T)g Fm(\\)15 b Fn(K)29 b Fr(=)c Fn(R)p Fr(.)e(Then)g(w)o(e)g(de\014ned)f(cardinals)g Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))25 b(=)0 941 y Fm(j)p Fn(L)48 923 y Fj(\003)71 941 y Fn(=T)132 923 y Fj(\003)154 941 y Fn(K)200 923 y Fj(\003)223 941 y Fm(j)17 b Fr(and)f Fn(f)5 b Fr(\()p Fn(T)h(=R)p Fr(\))15 b(=)e([)p Fn(T)6 b(=m)701 948 y Fk(T)746 941 y Fr(:)13 b Fn(R=m)880 948 y Fk(R)913 941 y Fr(].)0 1065 y Fp(Some)18 b(Remarks)f(on)h(E)h(and)g (F)100 1135 y Fr(F)l(rom)i(the)i(m)o(ultiplicativit)o(y)f(of)h(the)g (index)g(of)g(subgroups)d(and)i(of)h(the)g(degree)g(of)g(\014eld)f(ex-) 0 1205 y(tensions,)f(w)o(e)g(ha)o(v)o(e)h(the)g(follo)o(wing:)f(If)h Fn(M)28 b Fm(\033)23 b Fn(L)f Fr(is)f(an)h(extension)g(\014eld)f(of)h Fn(L)g Fr(and)f Fn(V)34 b Fm(\032)23 b Fn(M)28 b Fr(is)21 b(a)0 1274 y(v)m(aluation)14 b(ring)f(extending)g Fn(T)7 b Fr(,)14 b(then)g Fn(V)25 b Fr(ob)o(viously)13 b(extends)h Fn(R)g Fr(and)f Fn(e)p Fr(\()p Fn(V)6 b(=R)p Fr(\))15 b(=)f Fn(e)p Fr(\()p Fn(V)6 b(=T)h Fr(\))p Fn(e)p Fr(\()p Fn(T)f(=R)p Fr(\))15 b(and)0 1344 y Fn(f)5 b Fr(\()p Fn(V)h(=R)p Fr(\))15 b(=)f Fn(f)5 b Fr(\()p Fn(V)h(=T)h Fr(\))p Fn(f)e Fr(\()p Fn(T)h(=R)p Fr(\).)100 1414 y(Last)22 b(time)h(w)o(e)g(pro)o(v)o(ed)e(that)i(if)g([)p Fn(L)h Fr(:)g Fn(K)t Fr(])g Fn(<)g Fm(1)e Fr(then)h Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))p Fn(f)f Fr(\()p Fn(T)h(=R)p Fr(\))26 b Fm(\024)e Fr([)p Fn(L)g Fr(:)g Fn(K)t Fr(])e(and)g(in)0 1484 y(particuluar)13 b Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))14 b Fn(<)g Fm(1)h Fr(and)f Fn(f)5 b Fr(\()p Fn(T)h(=R)p Fr(\))15 b Fn(<)e Fm(1)p Fr(.)i(Later)g(it)g(will)f(arise)g(that)h(in)f (the)h(situations)f(w)o(e)g(are)0 1553 y(in)o(terested)i(in,)664 1582 y Fh(X)651 1688 y Fk(T)5 b Fj(7!)p Fk(R)758 1629 y Fn(e)p Fr(\()p Fn(T)h(=R)p Fr(\))p Fn(f)f Fr(\()p Fn(T)h(=R)p Fr(\))16 b(=)d([)p Fn(L)h Fr(:)f Fn(K)t Fr(])0 1771 y(where)j(the)h (sum)e(is)h(tak)o(en)h(o)o(v)o(er)f(all)g(v)m(aluations)g(rings)f Fn(T)24 b Fr(of)16 b(L)h(extending)f Fn(R)p Fr(.)0 1895 y Fp(Completions)100 1965 y Fr(Let)g Fn(R)h Fr(b)q(e)f(a)g(ring)g(and)f Fn(I)j Fm(\032)13 b Fn(R)k Fr(an)e(ideal)h(of)g Fn(R)p Fr(.)h(Then)e(w)o(e)h(de\014ne)g(the)g Fn(I)t Fr(-adic)f(completion)h (of)g Fn(R)p Fr(,)11 2023 y(^)0 2035 y Fn(R)p Fr(,)j(as)145 2023 y(^)134 2035 y Fn(R)e Fr(=)252 2013 y Fl(lim)259 2053 y Fj( )313 2035 y Fn(R=I)402 2017 y Fk(n)429 2035 y Fr(.)h(In)h(the)g(case)f(that)h Fn(R)g Fr(is)f(a)h(lo)q(cal)f(ring,)g (it)h(is)f(natural)g(to)g(tak)o(e)h Fn(I)j Fr(=)17 b Fn(m)1819 2042 y Fk(R)1870 2035 y Fr(and)0 2105 y(consider)g(the)i Fn(m)328 2112 y Fk(R)360 2105 y Fr(-adic)f(completion)g(of)g
Fn(R)p Fr(.)h(W)l(e)g(will)f(b)q(e)h(in)o(terested)e(in)h(the)h(the)g (case)f Fn(R)g Fm(\032)f Fn(K)22 b Fr(is)c(a)0 2175 y(discrete)e(v)m (aluation)g(ring)g(of)h(a)f(\014eld)g Fn(K)t Fr(.)0 2299 y Fp(Examples)i(of)g(Completions)100 2369 y Fn(R)c Fr(=)f Fn(k)r Fr([)p Fn(t)p Fr(])278 2378 y Fl(\()p Fk(t)p Fl(\))326 2369 y Fr(,)k(then)481 2357 y(^)470 2369 y Fn(R)522 2356 y Fm(\030)522 2372 y Fr(=)575 2369 y Fn(k)r Fr([[)p Fn(t)p Fr(]].)100 2448 y Fn(R)d Fr(=)f Fn(k)r Fr([)p Fn(t)p Fr(])278 2457 y Fl(\()p Fk(t)p Fj(\000)p Fk(\013)p Fl(\))400 2448 y Fr(for)j(some)g Fn(\013)e Fm(2)g Fn(k)r Fr(,)i(then)876 2435 y(^)865 2448 y Fn(R)917 2435 y Fm(\030)917 2450 y Fr(=)970 2448 y Fn(k)r Fr([[)p Fn(t)10 b Fm(\000)h Fn(\013)p Fr(]].)100 2526 y Fn(R)j Fr(=)f Fn(k)r Fr([)p Fn(t)264 2508 y Fj(\000)p Fl(1)317 2526 y Fr(])331 2536 y Fl(\()p Fk(t)362 2526 y Fi(\000)p Fd(1)408 2536 y Fl(\))426 2526 y Fr(,)j(then)581 2514 y(^)570 2526 y Fn(R)622 2513 y Fm(\030)622 2529 y Fr(=)675 2526 y Fn(k)r Fr([[)p Fn(t)749 2508 y Fj(\000)p Fl(1)801 2526 y Fr(]].)100 2600 y(If)k Fn(k)i Fm(6)p Fr(=)261 2586 y(\026)259 2600 y Fn(k)r Fr(,)e(and)f Fn(f)27 b Fr(is)19 b(a)h(separable,)f(irreducible)f(p)q (olynomial)h(o)o(v)o(er)h Fn(k)r Fr(,)f(with)h(deg)9 b Fn(f)26 b(>)20 b Fr(1,)g(tak)o(e)0 2670 y Fn(R)14 b Fr(=)g Fn(k)r Fr([)p Fn(t)p Fr(])179 2679 y Fl(\()p Fk(f)t Fl(\))235 2670 y Fr(.)i(Then)g(it)h(is)f(an)g(exercise)h(to)g(sho)o(w)e (that)1052 2657 y(^)1041 2670 y Fn(R)1093 2657 y Fm(\030)1093 2672 y Fr(=)1132 2677 y Fk(k)1170 2670 y Fn(l)q Fr([[)p Fn(t)p Fr(]],)h(where)g Fn(l)f Fr(=)e Fn(k)r Fr([)p Fn(t)p Fr(])p Fn(=)p Fr(\()p Fn(f)5 b Fr(\).)950 2790 y(15)p eop %%Page: 16 16 16 15 bop 100 50 a Fr(Here)17 b(w)o(e)h(ha)o(v)o(e)f(considered)f(the)h (completions)g(of)h(the)f(lo)q(cal)h(rings)e(of)i(the)g(function)f (\014eld)g Fn(k)r Fr(\()p Fn(t)p Fr(\).)0 120 y(The)e(completions)f(of) i(the)g(lo)q(cal)f(rings)f(of)i(the)g(functions)e(\014elds)h(of)h (other)f(curv)o(es)f(will)h(b)q(e)h(similar,)d(as)0 189 y(w)o(e)j(shall)g(see.)100 259 y(A)h(last)f(example:)h Fn(R)d Fr(=)g Fp(Z)598 268 y Fl(\()p Fk(p)p Fl(\))652 259 y Fr(.)j(Then)824 246 y(^)813 259 y Fn(R)e Fr(=)925 237 y Fl(lim)933 276 y Fj( )986 259 y Fp(Z)p Fn(=p)1071 241 y Fk(n)1098 259 y Fp(Z)g Fr(=)f Fp(Z)1236 266 y Fk(p)1259 259 y Fr(,)i(the)h(ring)f(of)h Fn(p)p Fr(-adic)f(in)o(tegers.)g(W)l(e)0 329 y(ha)o(v)o(e)g Fp(Z)150 336 y Fk(p)187 315 y Fm(\030)187 331 y Fr(=)239 329 y Fp(Z)p Fr([[)p Fn(t)p Fr(]])p Fn(=)p Fr(\()p Fn(t)11 b Fm(\000)g Fn(p)p Fr(\).)0 456 y Fp(More)19 b(Completions)100 527 y Fr(Again,)i(let)h Fn(R)h Fm(\032)g Fn(K)i Fr(b)q(e)e(a)e(discrete)h(v)m(aluation)f(ring.)g(Then)1330 515 y(^)1319 527 y Fn(R)h Fr(is)g(a)g(domain,)e(with)i(\014eld)f(of)0 598 y(fractions)218 586 y(^)205 598 y Fn(K)266 585 y Fm(\030)266 601 y Fr(=)320 598 y Fn(K)15 b Fm(\012)416 605 y Fk(R)471 586 y Fr(^)460 598 y Fn(R)p Fr(.)j(Tw)o(o)f(examples:)f (if)929 586 y(^)919 598 y Fn(R)f Fr(=)g Fn(k)r Fr([[)p Fn(t)p Fr(]],)h(then)1286 586 y(^)1272 598 y Fn(K)j Fr(=)c Fn(k)r Fr([[)p Fn(t)p Fr(]][)p Fn(t)1521 580 y Fj(\000)p Fl(1)1574 598 y Fr(])1603 585 y Fm(\030)1603 601 y Fr(=)1657 598 y Fn(k)r Fr(\(\()p Fn(t)p Fr(\)\).)j(And)f(if)11 660 y(^)0 672 y Fn(R)d Fr(=)g Fp(Z)140 679 y Fk(p)163 672 y Fr(,)i(then)h Fn(K)g Fr(=)d Fp(Q)462 679 y Fk(p)485 672 y Fr(.)100 748 y(No)o(w)21 b(w)o(e)f(giv)o(e)h(a)g(top)q(ological)g (construction)f(of)1070 736 y(^)1057 748 y Fn(K)t Fr(.)h
Fn(R)g Fr(is)g(a)g(discrete)f(v)m(aluation)h(ring,)f(so)g(w)o(e)0 818 y(ha)o(v)o(e)f(an)g(asso)q(ciated)g(v)m(aluation)g Fn(v)i Fr(:)d Fn(K)773 800 y Fj(\003)815 818 y Fm(!)g Fp(Z)p Fr(.)i(F)l(rom)e Fn(v)j Fr(w)o(e)e(construct)g(a)h(metric)f Fn(d)g Fr(on)g Fn(K)t Fr(:)h(for)f Fn(x)p Fr(,)0 888 y Fn(y)e Fm(2)d Fn(K)t Fr(,)j(de\014ne)f Fn(d)p Fr(\()p Fn(x;)8 b(y)r Fr(\))16 b(=)e(2)543 870 y Fj(\000)p Fk(v)q Fl(\()p Fk(x)p Fj(\000)p Fk(y)q Fl(\))704 888 y Fr(,)j(with)g(the)g (con)o(v)o(en)o(tion)e(that)j Fn(v)r Fr(\(0\))c(=)h Fm(1)p Fr(,)h(so)h(that)g Fn(d)p Fr(\()p Fn(x;)8 b(x)p Fr(\))16 b(=)e(0.)0 957 y(The)j(c)o(hoice)g(of)h(2)f(is)g(arbitrary;)e(an)o(y)i (p)q(ositiv)o(e)g(real)g(n)o(um)o(b)q(er)e(will)i(su\016ce.)g Fn(d)g Fr(giv)o(es)g(top)q(ology)g(on)g Fn(K)t Fr(,)0 1027 y(in)i(whic)o(h)f Fn(d)p Fr(\()p Fn(x;)8 b Fr(0\))21 b(is)e(small)36 b Fm(\()-8 b(\))36 b Fn(x)19 b Fm(2)g Fn(m)841 1009 y Fk(l)841 1041 y(R)893 1027 y Fr(for)g(some)f(large)h Fn(l)g Fm(2)g Fp(Z)p Fr(.)g(In)g(fact,)h(the)f(p)q(o)o(w)o(ers)f(of)i Fn(m)1918 1034 y Fk(R)0 1097 y Fr(form)c(a)g(basis)g(for)g(the)h(op)q (en)f(sets)g(con)o(taining)g(0.)100 1167 y(Then)g(the)i(top)q(ological) f(completion)f(of)i Fn(K)j Fr(with)c(resp)q(ect)g(to)h Fn(d)f Fr(has)g(a)g(natural)f(\014eld)h(structure)0 1238 y(whic)o(h)f(is)g(isomorphic)e(to)519 1225 y(^)506 1238 y Fn(K)t Fr(.)i(F)l(urthermore)e(the)j(closure)e(of)i Fn(R)g Fr(in)1306 1225 y(^)1293 1238 y Fn(K)j Fr(is)c(isomorphic)f(to) 1729 1225 y(^)1719 1238 y Fn(R)p Fr(.)100 1309 y(No)o(w)i(consider)g (the)h(\014eld)f(extension)g Fn(K)j Fm(\032)957 1297 y Fr(^)943 1309 y Fn(K)t Fr(.)e(Then)f(the)h(v)m(aluation)f(ring)1572 1297 y(^)1561 1309 y Fn(R)h Fr(extends)g Fn(R)p Fr(,)g(and)0 1381 y(w)o(e)e(ha)o(v)o(e)g Fn(e)p Fr(\()242 1368 y(^)230 1381 y Fn(R)q(=R)p Fr(\))e(=)g Fn(f)5 b Fr(\()478 1368 y(^)466 1381 y Fn(R)q(=R)p Fr(\))15 b(=)e(1.)0 1508 y Fp(A)19 b(Question)h(W)-5 b(as)19 b(Ask)n(ed)100 1579 y Fr(A)e(little)g(though)o(t)f(sho)o(ws)f(that)725 1567 y(^)711 1579 y Fn(K)771 1566 y Fm(\030)771 1582 y Fr(=)843 1557 y Fl(lim)854 1580 y Fi( )836 1608 y Fc(n)p Fi(2)p Fb(Z)917 1579 y Fn(K)q(=)p Fr(\()p Fn(m)1048 1586 y Fk(R)1080 1579 y Fr(\))1099 1561 y Fk(n)1127 1579 y Fr(,)i(and)f(that)1376 1567 y(^)1363 1579 y Fn(K)t(=)1444 1567 y Fr(^)1434 1579 y Fn(R)1486 1566 y Fm(\030)1486 1582 y Fr(=)1539 1579 y Fn(K)q(=R)p Fr(.)h(An)g(example:)0 1666 y(for)f Fn(R)e Fr(=)g Fp(Z)216 1675 y Fl(\()p Fk(p)p Fl(\))270 1666 y Fr(,)i Fp(Q)p Fn(=)p Fp(Z)403 1675 y Fl(\()p Fk(p)p Fl(\))471 1652 y Fm(\030)471 1668 y Fr(=)524 1666 y Fp(Z)p Fr([)p Fn(p)598 1648 y Fj(\000)p Fl(1)651 1666 y Fr(])p Fn(=)p Fp(Z)739 1652 y Fm(\030)739 1668 y Fr(=)792 1666 y Fm(f)p Fn(p)p Fr(-p)q(o)o(w)o(er)f(ro)q(ots)h(of)h(unit)o(y)f(in)g Fp(C)p Fm(g)p Fr(.)100 1735 y(W)l(e)g(ha)o(v)o(e)g(the)h(exact)h (sequence:)659 1860 y(0)13 b Fm(!)h Fp(Z)796 1867 y Fk(p)833 1860 y Fm(!)f Fp(Q)939 1867 y Fk(p)976 1860 y Fm(!)h Fp(Q)1083 1867 y Fk(p)1106 1860 y Fn(=)p Fp(Z)1166 1867 y Fk(p)1203 1860 y Fm(!)g Fr(0)0 1985 y(where)i Fp(Z)179 1992 y Fk(p)219 1985 y Fr(is)g(compact,)g Fp(Q)525 1992 y Fk(p)565 1985 y Fr(is)g(lo)q(cally)g(compact,)g(and)g Fp(Q)1125 1992 y Fk(p)1148 1985 y Fn(=)p Fp(Z)1208 1992 y Fk(p)1248 1985 y Fr(is)g(discrete.)100 2054 y(And)g(no)o(w)g(a)g (statemen)o(t)h(I)f(didn't)g(understand:)e(\\)p Fp(Q)1109 2061 y Fk(p)1149 2054 y Fr(is)i(self-dual)f(and)h Fn(\037)p Fr(\()p Fp(Z)1579 2061 y Fk(p)1602 2054 y Fr(\))f(=)e Fp(Q)1731 2061 y Fk(p)1754 2054 y Fn(=)p Fp(Z)1814 2061
y Fk(p)1837 2054 y Fr(".)0 2182 y Fp(Structure)19 b(of)f(a)h(Complete)h (D)n(VR)100 2252 y Fr(Let)c Fn(R)e Fm(\032)g Fn(K)20 b Fr(b)q(e)c(a)g(complete)g(discrete)g(v)m(aluation)g(ring)f(with)h(v)m (aluation)g Fn(v)r Fr(.)g(Then)f(c)o(ho)q(ose)h(a)g(set)0 2321 y Fn(B)j Fm(\032)e Fn(R)i Fr(suc)o(h)e(that)i(0)e Fm(2)g Fn(B)j Fr(and)e(the)h(natural)e(map)h Fn(B)h Fm(!)e Fn(R=m)1251 2328 y Fk(R)1302 2321 y Fr(is)h(a)g(bijection.)g(Also)g(c)o (ho)q(ose)g(for)0 2391 y(eac)o(h)f Fn(n)e Fm(2)g Fn(Z)21 b Fr(an)c(elemen)o(t)f Fn(\031)543 2398 y Fk(n)586 2391 y Fm(2)f Fn(K)21 b Fr(with)c Fn(v)r Fr(\()p Fn(\031)884 2398 y Fk(n)912 2391 y Fr(\))e(=)g Fn(n)p Fr(.)i(F)l(or)f(example,)h (tak)o(e)g Fn(\031)1497 2398 y Fk(n)1540 2391 y Fr(=)e Fn(\031)1624 2373 y Fk(n)1651 2391 y Fr(,)i(where)g Fn(\031)i Fr(is)e(a)0 2461 y(uniformizing)d(elemen)o(t)i(of)h Fn(R)p Fr(,)g(that)g(is)f Fn(m)801 2468 y Fk(R)847 2461 y Fr(=)e Fn(\031)r(R)p Fr(.)100 2531 y(Then)k(ev)o(ery)g(elemen)o(t)g(of)h Fn(R)f Fr(has)g(a)h(unique)e(expansion)h(as)1261 2493 y Fh(P)1313 2505 y Fj(1)1313 2545 y Fk(n)p Fl(=0)1399 2531 y Fn(b)1420 2538 y Fk(n)1448 2531 y Fn(\031)1476 2538 y Fk(n)1503 2531 y Fr(,)g(where)g Fn(b)1702 2538 y Fk(n)1747 2531 y Fm(2)f Fn(B)r Fr(,)i(and)0 2600 y(for)j(an)o(y)f(c)o (hoice)h(of)g(the)g Fn(b)509 2607 y Fk(n)536 2600 y Fr(,)g(that)g (series)f(con)o(v)o(erges)g(to)h(an)g(elemen)o(t)f(of)h Fn(R)p Fr(.)g(F)l(or)f Fn(K)26 b Fr(w)o(e)c(ha)o(v)o(e)f(the)0 2670 y(analagous)15 b(statemen)o(t)h(with)575 2633 y Fh(P)627 2645 y Fj(1)627 2685 y Fk(n)p Fl(=)p Fj(\000)p Fk(l)737 2670 y Fn(b)758 2677 y Fk(n)786 2670 y Fn(\031)814 2677 y Fk(n)858 2670 y Fr(for)g(some)g Fn(l)f Fm(\025)e Fr(0.)950 2790 y(16)p eop %%Page: 17 17 17 16 bop 100 50 a Fr(The)17 b(pro)q(of)g(of)g(this)g(is)g(straigh)o (tforw)o(ard:)e(for)i(example,)f(let)i Fn(r)f Fm(2)f Fn(R)p Fr(.)h(Then)g Fn(r)g Fr(=)e Fn(x)1674 57 y Fl(0)1697 50 y Fn(\031)1725 57 y Fl(0)1765 50 y Fr(for)i(some)0 120 y Fn(x)i Fm(2)f Fn(R)p Fr(,)h(and)f Fn(x)296 127 y Fl(0)337 120 y Fm(\021)g Fn(b)415 127 y Fl(0)457 120 y Fr(\(mo)q(d)h Fn(m)635 127 y Fk(R)667 120 y Fr(\))h(for)e(some)h Fn(b)933 127 y Fl(0)974 120 y Fm(2)f Fn(B)r Fr(.)h(That)g(is)g Fn(r)h Fr(=)d Fn(b)1397 127 y Fl(0)1420 120 y Fn(\031)1448 127 y Fl(0)1484 120 y Fr(+)12 b Fn(r)1557 127 y Fl(1)1599 120 y Fr(where)19 b Fn(r)1768 127 y Fl(1)1808 120 y Fm(2)g Fn(m)1904 127 y Fk(R)1936 120 y Fr(.)0 189 y(Again,)h Fn(r)185 196 y Fl(1)229 189 y Fr(=)h Fn(x)317 196 y Fl(1)340 189 y Fn(\031)368 196 y Fl(1)412 189 y Fr(for)f(some)g Fn(x)649 196 y Fl(1)693 189 y Fm(2)h Fn(R)p Fr(,)g(and)f Fn(x)949 196 y Fl(1)993 189 y Fm(\021)h Fn(b)1074 196 y Fl(1)1118 189 y Fr(\(mo)q(d)f Fn(m)1297 196 y Fk(R)1330 189 y Fr(\))h(for)g(some)f Fn(b)1601 196 y Fl(1)1645 189 y Fm(2)h Fn(B)r Fr(.)g(That)g(is,)0 259 y Fn(r)16 b Fr(=)d Fn(b)111 266 y Fl(0)134 259 y Fn(\031)162 266 y Fl(0)196 259 y Fr(+)e Fn(b)267 266 y Fl(1)289 259 y Fn(\031)317 266 y Fl(1)351 259 y Fr(+)g Fn(r)423 266 y Fl(2)446 259 y Fr(,)16 b(where)g Fn(r)642 266 y Fl(2)679 259 y Fm(2)e Fn(m)770 241 y Fl(2)770 273 y Fk(R)802 259 y Fr(.)j(Pro)q(ceeding)f(in)g(this)g(fashion,)f(w)o(e)i(ha)o(v)o(e)e Fn(r)h Fr(=)1708 222 y Fh(P)1760 234 y Fj(1)1760 274 y Fk(n)p Fl(=0)1846 259 y Fn(b)1867 266 y Fk(i)1884 259 y Fn(\031)1912 266 y Fk(i)1929 259 y Fr(.)0 384 y Fp(Finite)21 b(Extensions)e(of)f(Complete)i(D)n(VR's)0 509 y(Theorem)12
b Fr(Let)i Fn(R)g Fm(\032)g Fn(K)k Fr(b)q(e)c(a)f(complete)h(discrete)f (v)m(aluation)h(ring,)e(let)i Fn(L)g Fm(\033)g Fn(K)j Fr(b)q(e)d(a)g(\014eld)f(extension)0 579 y(of)23 b Fn(K)t Fr(,)f(and)f(let)i Fn(T)30 b Fr(b)q(e)22 b(a)h(v)m(aluation)f(ring)f (of)i Fn(L)f Fr(extending)g Fn(R)p Fr(.)h(Supp)q(ose)e(that)i Fn(e)g Fr(=)h Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))23 b(and)0 649 y Fn(f)e Fr(=)15 b Fn(f)5 b Fr(\()p Fn(T)h(=R)p Fr(\))19 b(are)e(\014nite.)g(Then)g([)p Fn(L)e Fr(:)h Fn(K)t Fr(])f(=)g Fn(ef)21 b(<)15 b Fm(1)p Fr(,)j Fn(T)24 b Fr(is)17 b(complete,)g Fn(T)25 b Fr(is)17 b(the)h(unique)e(v)m(aluation)0 718 y(ring)g(of)g Fn(L)h Fr(extending)f Fn(R)p Fr(,)h(and)f Fn(T)23 b Fr(is)16 b(the)h(in)o(tegral)f(closure)f(of)i Fn(R)g Fr(in)f Fn(L)p Fr(.)0 788 y Fp(Pro)r(of)k Fr(Cho)q(ose)i(a)f (set)h Fn(B)j Fm(\032)e Fn(R)f Fr(as)f(in)h(the)g(previous)f(section.)g (Also)h(c)o(ho)q(ose)f(a)h(set)g Fn(A)h Fm(\032)f Fn(T)29 b Fr(suc)o(h)0 858 y(that)19 b(the)g(natural)e(map)h Fn(A)f Fm(!)g Fn(T)6 b(=m)712 865 y Fk(T)761 858 y Fr(is)18 b(injectiv)o(e)h(and)e(has)h(image)g(a)g Fn(R=m)1498 865 y Fk(R)1531 858 y Fr(-basis)f(of)h Fn(T)6 b(=m)1834 865 y Fk(T)1865 858 y Fr(.)18 b(So)0 928 y Fm(j)p Fn(A)p Fm(j)c Fr(=)f Fn(f)5 b Fr(.)100 997 y(Let)23 b Fn(D)i Fr(=)f Fm(f)349 960 y Fh(P)401 1012 y Fk(a)p Fj(2)p Fk(A)490 997 y Fn(b)511 1004 y Fk(a)536 997 y Fn(a)p Fm(j)p Fn(b)597 1004 y Fk(a)645 997 y Fm(2)g Fn(B)r Fm(g)p Fr(.)f(Then)f Fn(D)j Fm(\032)f Fn(T)30 b Fr(and)21 b(the)i(natural)f(map)f Fn(D)26 b Fm(!)d Fn(T)6 b(=m)1863 1004 y Fk(T)1917 997 y Fr(is)0 1067 y(bijectiv)o(e.)100 1137 y(As)15 b Fn(e)g Fr(is)g(\014nite,)f Fn(T)23 b Fr(is)14 b(a)h(discrete)g(v)m(aluation)f (ring.)g(So)h(let)h Fn(\031)1217 1144 y Fk(T)1248 1137 y Fr(,)f Fn(\031)1305 1144 y Fk(R)1353 1137 y Fr(b)q(e)h(uniformizing)d (elemen)o(ts)h(of)0 1206 y Fn(T)21 b Fr(and)13 b Fn(R)p Fr(,)h(resp)q(ectiv)o(ely)l(.)g(Let)h Fn(v)600 1213 y Fk(T)631 1206 y Fr(,)f Fn(v)683 1213 y Fk(R)730 1206 y Fr(b)q(e)g(the)g(v)m(aluations)g(corresp)q(onding)d(to)k Fn(T)21 b Fr(and)13 b Fn(R)h Fr(resp)q(ectiv)o(ely)l(.)0 1276 y(Then)20 b Fn(v)158 1283 y Fk(T)189 1276 y Fr(\()p Fn(\031)236 1283 y Fk(R)270 1276 y Fr(\))g(=)g Fn(e)g Fr(b)o(y)g(the)h(de\014nition)e(of)h Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\).)21 b(So)f(for)g Fn(n)g Fr(=)g Fn(ei)14 b Fr(+)f Fn(j)s Fr(,)19 b(with)i(0)e Fm(\024)h Fn(j)j(<)c(e)p Fr(,)i(tak)o(e)0 1350 y Fn(\031)28 1357 y Fk(n)69 1350 y Fr(=)14 b Fn(\031)152 1331 y Fk(i)150 1363 y(R)183 1350 y Fn(\031)213 1326 y Fk(j)211 1364 y(T)243 1350 y Fr(;)i(w)o(e)h(ha)o(v)o(e)e Fn(v)485 1357 y Fk(T)517 1350 y Fr(\()p Fn(\031)564 1357 y Fk(n)592 1350 y Fr(\))f(=)g Fn(n)p Fr(.)100 1425 y(W)l(e)k(can)f(apply)g(the)h(discussion)d(from)i (the)h(previous)f(section)g(to)1383 1413 y(^)1373 1425 y Fn(T)7 b Fr(:)18 b(an)o(y)f Fn(t)e Fm(2)1628 1413 y Fr(^)1618 1425 y Fn(T)25 b Fr(has)17 b(a)h(unique)0 1495 y(represen)o(tation)23 b(as)h Fn(t)k Fr(=)515 1458 y Fh(P)568 1470 y Fj(1)568 1510 y Fk(n)p Fl(=0)654 1495 y Fn(d)680 1502 y Fk(n)707 1495 y Fn(\031)735 1502 y Fk(n)762 1495 y Fr(,)d(where)f Fn(d)979 1502 y Fk(n)1034 1495 y Fm(2)k Fn(D)q Fr(.)d(Then)f(write)h Fn(n)i Fr(=)h Fn(ei)17 b Fr(+)f Fn(j)27 b Fr(and)d(write)0 1565 y Fn(d)26 1572 y Fk(n)67 1565 y Fr(=)119 1527 y Fh(P)172 1580 y Fk(a)p Fj(2)p Fk(A)261 1565 y Fn(b)282 1572 y Fk(aij)339 1565 y Fn(a)p Fr(.)17 b(So)f(w)o(e)h(ha)o(v)o(e)394 1739 y Fn(t)d Fr(=)515 1675 y Fl(0)p Fj(\024)p Fk(i<)p Fj(1)547
1692 y Fh(X)478 1799 y Fl(0)p Fj(\024)p Fk(j)r(<e;a)p Fj(2)p Fk(A)696 1739 y Fn(b)717 1746 y Fk(aij)775 1739 y Fn(a\031)831 1718 y Fk(i)829 1751 y(R)862 1739 y Fn(\031)892 1715 y Fk(j)890 1754 y(T)936 1739 y Fr(=)1057 1692 y Fh(X)988 1799 y Fl(0)p Fj(\024)p Fk(j)r(<e;a)p Fj(2)p Fk(A)1198 1739 y Fr(\()1233 1677 y Fj(1)1217 1692 y Fh(X)1221 1798 y Fk(i)p Fl(=0)1298 1739 y Fn(b)1319 1746 y Fk(aij)1376 1739 y Fn(\031)1406 1718 y Fk(i)1404 1751 y(R)1437 1739 y Fr(\))p Fn(a\031)1512 1715 y Fk(j)1510 1754 y(T)1542 1739 y Fn(:)0 1921 y Fr(F)l(or)20 b(eac)o(h)h Fn(j)j Fr(and)d Fn(a)p Fr(,)g(the)h(inner)e(sum)g(is)h(an)g(elemen)o(t)g(of)h Fn(R)p Fr(.)f(Therefore)g(w)o(e)g(see)g(that)1722 1908 y(^)1712 1921 y Fn(T)29 b Fr(is)21 b(a)g(free)0 1992 y Fn(R)p Fr(-mo)q(dule)16 b(with)g(basis)f Fn(a\031)522 1968 y Fk(j)520 2006 y(T)569 1992 y Fr(and)h(rank)g Fn(e)p Fm(j)p Fn(A)p Fm(j)e Fr(=)g Fn(ef)5 b Fr(.)100 2061 y(W)l(e)16 b(will)h(complete)f(this)g(pro)q(of)g(next)h(time.)0 2186 y Fp(Lecture)j(8)c Fr(\(Septem)o(b)q(er)g(15,)h(1995\).)0 2256 y(Scrib)q(e:)f(Ben)h(Stein)o(b)q(erg)0 2381 y Fp(Theorem:)d Fr(Giv)o(en)i Fn(K)i Fm(\032)13 b Fn(L)j Fr(an)g(extension)g(of)g (\014elds,)f Fn(R)f Fm(\032)f Fn(K)20 b Fr(a)c(complete)g(v)m(aluation) g(ring,)f Fn(T)20 b Fm(\032)30 b Fn(L)0 2451 y Fr(a)17 b(v)m(aluation)f(ring)f(of)i Fn(L)g Fr(o)o(v)o(er)e Fn(R)i Fr(\()p Fn(T)h Fm(\\)12 b Fn(K)17 b Fr(=)d Fn(R)p Fr(\).)100 2560 y Fn(e)g(<)f Fm(1)p Fn(;)8 b(f)20 b(<)13 b Fm(1)p Fn(:)k Fr(Then:)100 2670 y(a\))g([)p Fn(L)c Fr(:)h Fn(K)t Fr(])f(=)h Fn(ef)5 b Fr(,)950 2790 y(17)p eop %%Page: 18 18 18 17 bop 100 50 a Fr(b\))17 b Fn(T)24 b Fr(is)17 b(a)h(complete)f(D)o (VR,)g Fn(T)24 b Fr(is)17 b(the)h(in)o(tegral)e(closure)g(of)i Fn(R)f Fr(in)g Fn(L)p Fr(,)g(and)g(the)g Fp(only)h Fr(v)m(aluation)0 120 y(ring)e(of)g Fn(L)h Fr(o)o(v)o(er)e Fn(R)p Fr(.)100 229 y Fp(Pro)r(of:)f Fr(Last)j(time:)f(W)l(e)h(sho)o(w)o(ed)e Fn(T)23 b Fr(is)16 b(a)h(D)o(VR,)1063 217 y(^)1053 229 y Fn(T)24 b Fr(is)16 b(free)g(of)h(rank)f Fn(ef)22 b Fr(as)16 b(an)h Fn(R)p Fr(-mo)q(dule)e(with)0 299 y(basis)g(tak)o(en)i (from)f Fn(T)7 b Fr(.)100 408 y(So)179 396 y(^)169 408 y Fn(T)21 b Fm(\032)13 b Fn(T)21 b Fm(\032)383 396 y Fr(^)374 408 y Fn(T)i Fr(so)16 b Fn(T)21 b Fr(=)600 396 y(^)590 408 y Fn(T)7 b Fr(.)100 518 y(So)15 b Fn(T)23 b Fr(is)15 b(free)g(of)h(rank)f Fn(ef)5 b Fr(.)17 b(Let)f Fn(\031)732 525 y Fk(R)781 518 y Fr(b)q(e)g(the)f(generator)g(of)h(the) g(maximal)e(ideal)h(of)h Fn(R)p Fr(.)g(Then)f Fn(T)1896 525 y Fk(\031)1919 530 y Fc(R)0 588 y Fr(is)j(also)f(free)h(of)g(rank)g Fn(ef)24 b Fr(o)o(v)o(er)17 b Fn(R)642 595 y Fk(\031)665 600 y Fc(R)696 588 y Fr(.)h(But)h Fn(L)d Fr(=)g Fn(T)963 595 y Fk(\031)986 600 y Fc(R)1017 588 y Fn(;)8 b(K)21 b Fr(=)16 b Fn(R)1195 595 y Fk(\031)1218 600 y Fc(R)1267 588 y Fr(\(since)i Fn(u\031)1469 570 y Fk(n)1467 602 y(T)1516 588 y Fr(for)g(some)f Fn(n)f(>)h Fr(0,)g Fn(u)h Fr(a)0 658 y(unit)e(of)h Fn(T)7 b Fr(\).)100 767 y(So)16 b([)p Fn(L)d Fr(:)h Fn(K)t Fr(])g(=)f Fn(ef)5 b Fr(.)100 877 y(No)o(w)15 b(since)h Fn(T)22 b Fr(is)16 b(\014nite)f(o)o(v)o(er)g Fn(R)h Fr(as)f(a)h(mo)q(dule,)f Fn(T)23 b Fr(is)15 b(in)o(tegral)f(o)o (v)o(er)h Fn(R)p Fr(.)h(So)g(the)g(in)o(tegral)e(closure)0 946 y(or)h Fn(R)h Fr(in)g Fn(L)g Fr(equals)f(the)h(in)o(tegral)f (closure)f(of)i Fn(T)23 b Fr(in)16 b Fn(L)p Fr(.)f(But)i(this)e(is)g Fn(T)23 b Fr(since)15 b Fn(T)23 b Fr(is)16 b(a)g(v)m(aluation)f(ring)g (of)0 1016 y Fn(L)i Fr(and)e(hence)i(in)o(tegrally)e(closed.)100
1126 y(So)j Fn(T)24 b Fr(=)18 b(the)g(in)o(tegral)g(closure)f(of)h Fn(R)g Fm(\032)g Fr(an)o(y)g(non-trivial)e(v)m(aluation)j(ring)e(of)h Fn(L)h Fr(extending)f Fn(R)p Fr(.)0 1196 y(But)f Fn(T)24 b Fr(is)16 b(a)g(maximal)g(v)m(aluation)g(ring)f(in)h Fn(L)h Fr(since)f Fn(T)24 b Fr(is)16 b(discrete.)100 1305 y Fp(Corollary:)k Fr(If)f Fn(R)h Fr(is)f(a)g(complete)g(D)o(VR)g (and)g Fn(L)f Fm(\033)g Fn(K)24 b Fr(is)19 b(an)o(y)f(algebraic)h (extension,)g(then)g Fn(R)0 1375 y Fr(has)d(exactly)h(one)g(extension)f (to)h(a)f(v)m(aluation)h(ring)e(of)i Fn(L)p Fr(.)100 1484 y Fp(Pro)r(of:)d Fr(W)l(e)i(kno)o(w)g(from)f(an)g(old)h(result)f (that)i(there)e(is)h(at)g(least)g(one,)g(so)g(let's)f(sho)o(w)g (there's)h(at)0 1554 y(most)j(one.)h(It)g(su\016ces)f(to)h(consider)e (the)i(case)g([)p Fn(L)f Fr(:)g Fn(K)t Fr(])g(since)h(t)o(w)o(o)f (distinct)g(suc)o(h)g(v)m(aluation)g(rings)0 1624 y(will)d(di\013er)g (in)g(a)h(\014nite)f(extension.)100 1734 y(Let)h Fn(T)218 1741 y Fl(1)240 1734 y Fn(;)8 b(T)291 1741 y Fl(2)331 1734 y Fr(b)q(e)16 b(t)o(w)o(o)g(suc)o(h.)g(But)h Fn(e)741 1741 y Fk(i)757 1734 y Fn(f)781 1741 y Fk(i)812 1734 y Fm(\024)d Fr([)p Fn(L)f Fr(:)h Fn(K)t Fr(])g Fm(\024)f(1)k Fr(so)f(b)o(y)g(the)h(theorem)f Fn(T)1586 1741 y Fl(1)1622 1734 y Fr(=)d Fn(T)1703 1741 y Fl(2)1726 1734 y Fr(.)100 1843 y(So)j(in)g(particular)f(w)o(e)h(ha)o(v)o(e)g(the)h(follo)o(wing)e (diagrams;)100 1953 y(Let)j Fn(L)g Fr(b)q(e)g(\014nite)f(o)o(v)o(er)g Fn(K)t Fr(.)h(The)g(w)o(e)f(get)h Fn(K)933 1935 y Fj(\003)972 1953 y Fm(!)e Fp(Z)i Fr(and)f Fn(L)1223 1935 y Fj(\003)1262 1953 y Fm(!)e Fr(1)p Fn(=e)p Fp(Z)j Fr(with)g(the)g(ob)o(vious)e(maps)0 2022 y(b)q(et)o(w)o(een)g(these)h(arro)o(ws.)100 2132 y(Th)o(us)e(w)o(e)h(ha)o(v)o(e)g(a)h(surjection)e Fn(K)730 2114 y Fk(alg)q Fj(\003)822 2132 y Fm(!)f Fp(Q)p Fr(.)100 2242 y Fp(Exercise:)h Fr(Sho)o(w)e Fn(L)515 2228 y Fm(\030)515 2244 y Fr(=)568 2242 y Fn(K)614 2224 y Fk(n)655 2242 y Fr(where)h Fn(L)g Fr(is)g(giv)o(en)g(the)h(v)m(aluation)f(top)q (ology)h(and)e Fn(K)1663 2224 y Fk(n)1705 2242 y Fr(is)h(giv)o(en)g (the)0 2311 y(pro)q(duct)i(top)q(ology)h(\([)p Fn(L)d Fr(:)f Fn(K)t Fr(])h(=)f Fn(n)p Fr(\).)k(This)f(is)g(also)g(true)g(for) g Fn(T)24 b Fr(o)o(v)o(er)16 b Fn(R)p Fr(.)100 2421 y Fp(Hin)n(t:)k Fr(Sho)o(w)e(that)h(the)g(top)q(ology)g(on)g Fn(K)k Fr(is)18 b(indep)q(enden)o(t)g(of)h(the)g(c)o(hoice)g(of)g (bases,)f(then)h(use)0 2491 y(the)e(bases)e(from)h(the)h(pro)q(of)f(of) h(the)g(theorem.)100 2600 y Fp(Theorem:)i Fr(Let)i Fn(R)g Fr(b)q(e)g(a)g(D)o(VR)g(of)g(a)g(\014eld)f Fn(K)t Fr(,)h(with)f (completion)1453 2588 y(^)1439 2600 y Fn(K)t Fr(\(=)h Fn(K)1619 2563 y Fh(N)1674 2615 y Fk(R)1726 2588 y Fr(^)1715 2600 y Fn(R)p Fr(\))g(and)g(let)0 2670 y Fn(L)14 b Fm(\033)f Fn(K)21 b Fr(b)q(e)c(an)o(y)f(\014nite)g(\014eld)g(extension.)g(Then)g (there)h(exists)f(bijections)g(b)q(et)o(w)o(een)g(the)h(sets;)950 2790 y(18)p eop %%Page: 19 19 19 18 bop 100 50 a Fn(A)17 b Fr(=)f Fm(f)p Fn(T)7 b Fm(j)p Fn(T)23 b Fr(is)16 b(a)h(v)m(aluation)f(ring)g(of)g Fn(L)h Fr(extending)f Fn(R)p Fm(g)100 163 y Fn(B)j Fr(=)d Fm(f)p Fr(\014eld)g(em)o(b)q(eddings)e(o)o(v)o(er)i Fn(K)k Fr(of)d Fn(L)d Fm(!)967 151 y Fr(^)953 163 y Fn(K)999 145 y Fk(alg)1057 163 y Fm(g)p Fn(=)p Fm(f)p Fr(Aut)1243 170 y Fl(^)1233 179 y Fk(K)1284 151 y Fr(^)1271 163 y Fn(K)1317 145 y Fk(alg)1374 163 y Fm(g)100 280 y Fn(C)19 b Fr(=)e Fn(S)s(pec)p Fr(\()p Fn(L)376 243 y Fh(N)431 295 y Fk(K)491 268 y Fr(^)478 280 y Fn(K)s Fr(\))100 396 y Fp(Examples:)e Fn(L)f Fr(=)g Fn(K)t Fr([)p Fn(x)p Fr(])p Fn(=)p Fr(\()p
Fn(f)5 b Fr(\).)17 b(Then)f Fn(B)h Fr(=)c Fm(f)p Fr(zero)q(es)k(of)f(f) h(in)1268 384 y(^)1255 396 y Fn(K)1301 378 y Fk(alg)1359 396 y Fn(=)c Fm(\030)h Fn(R)p Fm(g)p Fr(.)100 510 y(But)j Fn(S)s(pec)p Fr(\()p Fn(L)364 473 y Fh(N)419 525 y Fk(K)479 497 y Fr(^)465 510 y Fn(K)t Fr(\))544 497 y Fm(\030)544 513 y Fr(=)610 497 y(^)597 510 y Fn(K)t Fr([)p Fn(x)p Fr(])p Fn(=)p Fr(\()p Fn(f)5 b Fr(\))100 626 y(whic)o(h)15 b(are)h(the)h(monic)f(irreducible)e(factors)j(of)f Fn(f)23 b Fr(in)1141 614 y(^)1128 626 y Fn(K)t Fr([)p Fn(x)p Fr(])p Fn(=)p Fr(\()p Fn(f)5 b Fr(\).)100 736 y(This)17 b(theorem)h(is)g(analagous)f(to)i(the)f(fact)i(pro)o(v)o(ed)c(b)o(y)j (Kummer)d(that)j(for)f(most)g(primes)f Fn(p)h Fr(in)0 805 y Fp(Z)e Fr(in)g Fp(Q)p Fr([)p Fn(x)p Fr(])p Fn(=)p Fr(\()p Fn(f)5 b Fr(\))17 b(primes)d(lying)i(ab)q(o)o(v)o(e)g Fn(p)g Fr(corresp)q(ond)e(to)i(irreducible)e(factors)i(of)g Fn(f)22 b Fr(mo)q(d)15 b Fn(p)h Fr(in)g Fp(F)1851 812 y Fg(p)1878 805 y Fr([)p Fp(x)p Fr(].)100 921 y Fp(Pro)r(of:)f Fr(Giv)o(en)j Fn(T)k Fm(2)16 b Fn(A)p Fr(,)i(\014rst)f(note)g Fn(T)25 b Fr(is)17 b(a)h(D)o(VR.)f(Then)g Fn(L)e Fm(\032)h Fn(L)1374 908 y Fr(^)1361 921 y Fn(K)21 b Fr(in)1489 908 y(^)1483 921 y Fn(L)16 b Fr(=)f Fn(L)1629 883 y Fh(N)1684 936 y Fk(T)1734 908 y Fr(^)1724 921 y Fn(T)23 b(,)-8 b Fm(!)1860 908 y Fr(^)1847 921 y Fn(K)1893 903 y Fk(alg)0 997 y Fr(o)o(v)o(er)16 b Fn(K)t Fr(.)g(Note)h(in)f(fact)i Fn(L)507 984 y Fr(^)494 997 y Fn(K)f Fr(=)612 984 y(^)606 997 y Fn(L)f Fr(since)g Fn(L)826 984 y Fr(^)812 997 y Fn(K)21 b Fr(is)16 b(\014nite)g(o)o(v)o(er)g Fn(K)t Fr(.)100 1107 y(Let)f Fn(')f Fm(2)g Fn(B)r Fr(,)h(then)f Fn(')g Fr(:)f Fn(L)h(,)-8 b Fm(!)666 1094 y Fr(^)652 1107 y Fn(K)698 1089 y Fk(alg)756 1107 y Fr(.)14 b(Also)h Fn(id)f Fr(:)993 1094 y(^)980 1107 y Fn(K)j(,)-8 b Fm(!)1122 1094 y Fr(^)1109 1107 y Fn(K)18 b Fr(agree)d(on)f Fn(K)t Fr(.)h(So)f Fn(')1548 1069 y Fh(N)1612 1107 y Fn(id)g Fr(:)f Fn(L)1738 1069 y Fh(N)1794 1122 y Fk(K)1854 1094 y Fr(^)1840 1107 y Fn(K)18 b Fm(!)13 1170 y Fr(^)0 1183 y Fn(K)46 1165 y Fk(alg)103 1183 y Fr(.)f(The)f(k)o(ernel)g(of)h(this)f (map)g(is)g(in)g Fn(C)100 1294 y Fr(Let)h Fn(})d Fm(2)g Fn(C)t Fr(.)h(Then)i(since)f Fn(L)d Fm(\032)h Fn(L)746 1256 y Fh(N)802 1309 y Fk(K)862 1281 y Fr(^)848 1294 y Fn(K)k Fm(!)c Fr(\()p Fn(L)1033 1256 y Fh(N)1089 1309 y Fk(K)1149 1281 y Fr(^)1136 1294 y Fn(K)s Fr(\))p Fn(=})j Fr(a)g(domain)e(\014nite)h(o)o(v)o(er)1740 1281 y(^)1727 1294 y Fn(K)t Fr(,)g(a)h(\014eld.)0 1370 y(So)f Fn(})g Fr(is)h(maximal)e(and)h(hence)g(w)o(e)h(can)f(extend)h(this)f(map)g(to) g(an)h(em)o(b)q(edding)e(in)o(to)1637 1357 y(^)1624 1370 y Fn(K)1670 1352 y Fk(alg)1727 1370 y Fr(.)100 1481 y(Finally)l(,)j(if) h Fn(')f Fm(2)g Fn(B)r Fr(,)i(i.e.)f Fn(')f Fr(:)g Fn(L)g(,)-8 b Fm(!)811 1468 y Fr(^)798 1481 y Fn(K)844 1463 y Fk(alg)921 1481 y Fr(then)19 b(tak)o(e)h Fn(')1181 1463 y Fj(\000)p Fl(1)1253 1481 y Fr(of)g(the)f(unique)g(extension)g(of)1859 1468 y(^)1848 1481 y Fn(R)h Fr(to)13 1539 y(^)0 1552 y Fn(K)46 1534 y Fk(alg)103 1552 y Fr(.)100 1661 y Fp(Exercise:)d Fr(Sho)o(w)f(the)g(appropriate)f(maps)g(are)i(in)o(v)o(erses.)100 1771 y(Note)g(since)f(the)h(second)f(set)g(is)g(\014nite,)h(all)f (these)g(sets)h(are)f(\014nite.)100 1882 y Fp(Next)21 b(Time:)d Fr(W)l(e'll)f(sho)o(w)661 1844 y Fh(Q)708 1882 y Fr(\(\()p Fn(L)788 1844 y Fh(N)863 1869 y Fr(^)853 1882 y Fn(T)7 b Fr(\))925 1868 y Fm(\030)925 1884 y Fr(=)980 1882 y(\()p Fn(L)1041 1844 y Fh(N)1097 1897 y Fk(K)1157
1869 y Fr(^)1143 1882 y Fn(K)t Fr(\))p Fn(=)1233 1841 y Fm(p)p 1275 1841 25 2 v 41 x Fr(0)18 b(where)f Fn(T)26 b Fr(runs)16 b(o)o(v)o(er)h(the)h(ab)q(o)o(v)o(e)0 1951 y(set)f(and)175 1914 y Fh(P)236 1951 y Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))p Fn(f)f Fr(\()p Fn(T)h(=R)p Fr(\))15 b Fm(\024)f Fr([)p Fn(L)f Fr(:)h Fn(K)t Fr(].)0 2078 y Fp(Lecture)20 b(9)c Fr(\(Septem)o(b)q(er)g(18,)h(1995\).)0 2147 y(Scrib)q(e:)f(Emiliano)f(G\023)-25 b(omez.)100 2273 y(W)l(e)16 b(w)o(ere)g(pro)o(ving)f(the)i(follo)o(wing)f(theorem:) 100 2383 y Fp(Theorem:)10 b Fr(Let)j Fn(R)g Fr(b)q(e)f(a)g(discrete)g (v)m(aluation)g(ring)f(of)i(a)f(\014eld)g Fn(K)t Fr(,)g(and)f Fn(L)h Fr(a)h(\014nite)f(\014eld)f(extension)0 2453 y(of)17 b Fn(K)t Fr(.)f(Then)g(w)o(e)g(ha)o(v)o(e)g(bijections)g(\(see)h(last)g (lecture\))f(b)q(et)o(w)o(een)h(the)g(follo)o(wing)e(three)h(sets:)100 2523 y Fm(f)p Fn(T)k Fm(j)14 b Fn(T)24 b Fr(is)16 b(a)g(v)m(aluation)h (ring)e(of)i Fn(L)f Fr(extending)h Fn(R)p Fr(,)f(i.e.)h Fn(T)h Fm(\\)11 b Fn(K)18 b Fr(=)13 b Fn(R)p Fm(g)100 2596 y(f)p Fn(K)t Fr(-em)o(b)q(eddings)g Fn(L)h Fm(\000)-8 b(!)614 2584 y Fr(^)600 2596 y Fn(K)646 2578 y Fk(alg)704 2596 y Fm(g)p Fn(=)13 b Fm(\030)833 2584 y Fr(^)820 2596 y Fn(K)100 2670 y Fr(Sp)q(ec\()p Fn(L)262 2633 y Fh(N)318 2685 y Fk(K)378 2657 y Fr(^)364 2670 y Fn(K)t Fr(\))950 2790 y(19)p eop %%Page: 20 20 20 19 bop 100 50 a Fr(Also,)16 b(these)g(sets)h(are)f(\014nite,)g(and) 772 12 y Fh(Q)819 65 y Fk(T)865 37 y Fr(^)859 50 y Fn(L)893 32 y Fk(T)938 37 y Fm(\030)938 52 y Fr(=)987 57 y Fl(^)977 66 y Fk(K)1029 50 y Fr(\()p Fn(L)1090 12 y Fh(N)1146 65 y Fk(K)1206 37 y Fr(^)1192 50 y Fn(K)t Fr(\))p Fn(=)1282 7 y Fh(p)p 1332 7 64 2 v 43 x Fr(\(0\))q(.)g(Also,)100 88 y Fh(P)152 140 y Fk(T)192 125 y Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))p Fn(f)f Fr(\()p Fn(T)h(=R)p Fr(\))19 b Fm(\024)e Fr([)p Fn(L)g Fr(:)g Fn(K)t Fr(],)h(and)g(if)g Fn(L)g Fm(\033)e Fn(K)23 b Fr(is)18 b(separable,)f(then)1499 82 y Fh(p)p 1549 82 V 43 x Fr(\(0\))h(=)f(\(0\),)i(and)f(the)0 195 y(ab)q(o)o(v)o(e)e(inequalit)o(y)g(b)q(ecomes)g(an)h(equalit)o(y)l (.)100 304 y Fp(Pro)r(of)d Fr(of)j(the)g(last)f(statemen)o(ts)g(\(the)h (bijections)f(are)h(left)g(for)f(us)g(to)h(pro)o(v)o(e\):)100 374 y(The)f(middle)f(set)i(is)f(\014nite)g(b)q(ecause)h([)p Fn(L)c Fr(:)h Fn(K)t Fr(])i(is)g(\014nite.)h(Another)f(pro)q(of)g(of)h (the)g(\014niteness)e(is:)100 446 y(dim)193 453 y Fl(^)184 462 y Fk(K)221 446 y Fn(L)263 409 y Fh(N)319 461 y Fk(K)379 434 y Fr(^)365 446 y Fn(K)28 b Fr(=)22 b(dim)580 453 y Fk(K)617 446 y Fn(L)i Fr(=)f([)p Fn(L)g Fr(:)g Fn(K)t Fr(])g Fn(<)h Fm(1)p Fr(,)e(so)f Fn(L)1185 409 y Fh(N)1241 461 y Fk(K)1301 434 y Fr(^)1288 446 y Fn(K)26 b Fr(satis\014es)21 b(the)i(ascending)e(and)0 516 y(descending)16 b(c)o(hain)h(conditions)f (on)i(ideals.)e(It)i(is)g(an)f(Artin)h(ring)e(\(see)i(c)o(h.8)f(of)h(A) o(tiy)o(ah)g(&)f(MacDon-)0 588 y(ald\).)22 b(Th)o(us)f(Sp)q(ec\()p Fn(L)416 551 y Fh(N)472 603 y Fk(K)532 576 y Fr(^)518 588 y Fn(K)t Fr(\))i(is)f(\014nite.)f(Next,)j(\()p Fn(L)1011 551 y Fh(N)1067 603 y Fk(K)1127 576 y Fr(^)1113 588 y Fn(K)t Fr(\))p Fn(=)1203 546 y Fh(p)p 1253 546 V 42 x Fr(\(0\))g(=)f(\()p Fn(L)1463 551 y Fh(N)1519 603 y Fk(K)1579 576 y Fr(^)1566 588 y Fn(K)t Fr(\))p Fn(=)1664 551 y Fh(T)1706 603 y Fk(\045)16 b(pr)q(ime)1862 588 y Fn(\045)1911 575 y Fm(\030)1911 591 y Fr(=)0 633 y Fh(Q)47 685 y Fk(\045)70 670 y Fr(\()p Fn(L)131 633 y Fh(N)187 685 y Fk(K)247 657 y Fr(^)234 670 y Fn(K)t Fr(\))p Fn(=\045)e Fr(=)416
633 y Fh(Q)463 685 y Fk(T)509 657 y Fr(^)503 670 y Fn(L)537 652 y Fk(T)585 670 y Fr(.)100 739 y(Here)e(the)h(\014rst)e(isomorphism) e(is)j(a)g(consequence)g(of)h(the)f(Chinese)g(Remainder)e(Theorem,)h (whic)o(h)0 809 y(applies)f(b)q(ecause)h(Artin)g(rings)f(ha)o(v)o(e)g (dimension)f(zero,)i(and)f(so)h(ev)o(ery)g(prime)f(ideal)h(is)g (maximal.)f(Hence)0 878 y(an)o(y)16 b(t)o(w)o(o)g(distinct)g(prime)g (ideals)f(m)o(ust)h(b)q(e)g(coprime.)100 948 y(If)j Fn(L)g Fr(is)g(separable)e(o)o(v)o(er)h Fn(K)t Fr(,)h(then)g Fn(L)835 935 y Fm(\030)835 951 y Fr(=)892 948 y Fn(K)t Fr([)p Fn(X)t Fr(])p Fn(=)p Fr(\()p Fn(f)5 b Fr(\),)20 b(where)f Fn(f)25 b Fr(is)18 b(irreducible)f(and)i(separable.)0 1018 y(So)d(w)o(e)h(ha)o(v)o(e)100 1088 y Fn(L)142 1050 y Fh(N)197 1103 y Fk(K)257 1075 y Fr(^)244 1088 y Fn(K)308 1074 y Fm(\030)308 1090 y Fr(=)378 1075 y(^)365 1088 y Fn(K)s Fr([)p Fn(X)t Fr(])p Fn(=)p Fr(\()p Fn(f)5 b Fr(\))595 1074 y Fm(\030)595 1090 y Fr(=)651 1050 y Fh(Q)698 1103 y Fk(g)743 1075 y Fr(^)730 1088 y Fn(K)t Fr([)p Fn(X)t Fr(])p Fn(=)p Fr(\()p Fn(g)r Fr(\),)19 b(the)g(pro)q(duct)f(tak) o(en)i(o)o(v)o(er)e(monic)g(p)q(olynomials)f Fn(g)0 1169 y Fr(of)70 1157 y(^)57 1169 y Fn(K)t Fr([)p Fn(X)t Fr(])f(whic)o(h)g (divide)g Fn(f)5 b Fr(.)17 b(W)l(e)g(see)f(in)g(this)g(case)h(that)1078 1127 y Fh(p)p 1128 1127 V 42 x Fr(\(0\))d(=)g(\(0\).)j(Next,)100 1244 y Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))p Fn(f)f Fr(\()p Fn(T)h(=R)p Fr(\))23 b(=)f Fn(e)p Fr(\()560 1232 y(^)550 1244 y Fn(T)8 b(=)622 1232 y Fr(^)612 1244 y Fn(R)p Fr(\))p Fn(f)d Fr(\()728 1232 y(^)717 1244 y Fn(T)j(=)790 1232 y Fr(^)779 1244 y Fn(R)p Fr(\))23 b(=)e([)939 1232 y(^)933 1244 y Fn(L)h Fr(:)1038 1232 y(^)1025 1244 y Fn(K)t Fr(].)f(So)g(the)g (desired)g(inequalit)o(y)f(no)o(w)h(follo)o(ws)0 1314 y(from)100 1346 y Fh(P)152 1399 y Fk(T)184 1384 y Fr([)204 1371 y(^)198 1384 y Fn(L)f Fr(:)299 1371 y(^)286 1384 y Fn(K)t Fr(])g Fm(\024)g Fr([)p Fn(L)g Fr(:)h Fn(K)t Fr(].)f(Here)h(w)o(e)f(sum)f(o)o(v)o(er)h(the)h(completions)e(of)i Fn(L)f Fr(with)g(resp)q(ect)h(to)g(the)0 1454 y(distinct)16 b(v)m(aluation)g(rings)g Fn(T)23 b Fr(extending)17 b Fn(R)p Fr(.)f(This)g(completes)g(the)h(pro)q(of.)100 1563 y Fp(Remark:)23 b Fr(In)h(fact,)i(one)e(has)726 1526 y Fh(P)779 1578 y Fk(T)819 1563 y Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))p Fn(f)f Fr(\()p Fn(T)h(=R)p Fr(\))p Fn(g)r Fr(\()p Fn(T)g(=R)p Fr(\),)26 b(where)f Fn(g)r Fr(\()p Fn(T)6 b(=R)p Fr(\))27 b(=)e(lengh)o(t)f(of)0 1638 y(\()p Fn(L)61 1601 y Fh(N)117 1653 y Fk(K)177 1625 y Fr(^)164 1638 y Fn(K)s Fr(\))228 1645 y Fk(\045)266 1638 y Fr(as)13 b(a)h(mo)q(dule)f(o)o(v)o(er)g(itself,)h(if)g Fn(T)21 b Fm($)14 b Fn(\045)p Fr(.)f(Here)h(the)h Fn(\045)p Fr('s)e(are)g(the)h(prime)f(ideals)g(of)h Fn(L)1788 1601 y Fh(N)1844 1653 y Fk(K)1904 1625 y Fr(^)1890 1638 y Fn(K)t Fr(.)0 1708 y(The)g(latter,)g(b)q(eing)f(an)h(Artin)f(ring,)g (is)g(the)h(pro)q(duct)g(of)g(its)f(prime)g(lo)q(calizations.)g Fn(g)r Fr(\()p Fn(T)6 b(=R)p Fr(\))14 b(is)f(a)h(p)q(o)o(w)o(er)0 1777 y(of)20 b Fn(p)p Fr(,)h(where)f Fn(p)g Fr(=)g(c)o(har)p Fn(K)t Fr(.)f(Note)i(that)f(in)g(the)h(case)f(where)g Fn(p)g Fr(=)g(0)p Fn(;)28 b(L)20 b Fr(:)g Fn(K)k Fr(is)c(separable,)f (and)g(so)0 1847 y Fn(g)r Fr(\()p Fn(T)6 b(=R)p Fr(\))15 b(=)g(1)i(agrees)g(with)g(the)g(theorem.)g(W)l(e)g(will)g(study)g (function)g(\014elds,)f(and)h(it)g(turns)f(out)i(that)0 1917 y(w)o(e)e(will)g(alw)o(a)o(ys)g(ha)o(v)o(e)g Fn(g)r Fr(\()p Fn(T)6 b(=R)p Fr(\))14 b(=)f(1.)j(So)h(w)o(e)f(will)g(not)h (deal)f(with)g Fn(g)r Fr(\()p Fn(T)6 b(=R)p Fr(\))17
b(an)o(ymore.)100 2027 y(Let)d(us)f(consider)f(function)h(\014elds)f (no)o(w:)h(Let)h Fn(K)j Fr(b)q(e)d(\014nitely)f(generated)g(of)h (trascendence)e(degree)0 2096 y(1)17 b(o)o(v)o(er)e Fn(k)r Fr(.)h(Then)g(for)g(ev)o(ery)h Fn(z)f Fm(2)e Fn(K)676 2078 y Fj(\003)716 2096 y Fr(one)i(has)100 2129 y Fh(P)152 2181 y Fk(R)193 2166 y Fn(v)217 2173 y Fk(R)250 2166 y Fr(\()p Fn(z)r Fr(\)[)p Fn(R=m)p 390 2174 44 2 v 435 2178 a Fk(R)487 2166 y Fr(:)k Fn(k)r Fr(])f(=)g(0,)h(where)g(the)g(sum) f(is)g(tak)o(en)i(o)o(v)o(er)e(the)h(discrete)g(v)m(aluation)f(rings)0 2236 y Fn(R)25 b Fr(of)f Fn(K)k Fr(o)o(v)o(er)c Fn(k)r Fr(,)f(where)h Fn(m)p 529 2244 V 12 x Fk(R)630 2236 y Fr(is)g(the)g(maximal)g(ideal)f(of)i Fn(R)p Fr(,)f(and)g(where)g Fn(v)1541 2243 y Fk(R)1598 2236 y Fr(is)g(the)g(v)m(aluation)0 2311 y Fn(v)24 2318 y Fk(R)71 2311 y Fr(:)13 b Fn(K)144 2293 y Fj(\003)181 2311 y Fm(\000)-9 b(!)14 b Fn(K)321 2293 y Fj(\003)344 2311 y Fn(=R)407 2293 y Fj(\003)468 2283 y(')444 2311 y Fm(\000)-9 b(!)14 b(Z)t Fr(.)100 2421 y(Let)g(us)g(\\almost")f(pro)o(v)o(e)g(this,)h(using)f(the)h (inequalit)o(y)g(of)g(the)g(ab)q(o)o(v)o(e)g(theorem.)f(W)l(e)i(could)e (pro)o(v)o(e)0 2491 y(it)h(if)g(w)o(e)g(knew)g(that)g(w)o(e)g(actually) g(ha)o(v)o(e)f(an)h(equalit)o(y)l(.)f(In)h(fact)g(w)o(e)g(do,)g(as)f(w) o(e)h(will)f(sho)o(w)g(later.)h(T)l(aking)0 2560 y(that)j(for)f(gran)o (ted)g(no)o(w,)f(w)o(e)i(can)f(pro)o(v)o(e)f(our)h(assertion:)100 2670 y(If)h Fn(z)i Fr(is)d(algebraic)g(o)o(v)o(er)g Fn(k)r Fr(,)g(then)h(it)g(will)f(b)q(e)h(a)g(unit)f(in)g Fn(R)h Fr(\(i.e.)g Fn(z)g Fm(2)d Fn(R)1429 2652 y Fj(\003)1452 2670 y Fr(\),)j(and)f(so)h Fn(v)1685 2677 y Fk(R)1717 2670 y Fr(\()p Fn(z)r Fr(\))f(=)e(0)i(for)950 2790 y(20)p eop %%Page: 21 21 21 20 bop 0 50 a Fr(eac)o(h)13 b Fn(R)p Fr(.)g(So)g(let)h(us)e(assume)g (that)i Fn(z)i Fr(is)c(not)i(algebraic)e(o)o(v)o(er)g Fn(k)r Fr(.)h(Then)g Fn(K)k Fr(is)c(a)g(\014nite)g(extension)g(of)h Fn(k)r Fr(\()p Fn(z)r Fr(\),)0 120 y(and)i(eac)o(h)h(discrete)f(v)m (aluation)h(ring)f Fn(R)h Fr(of)h Fn(K)i Fr(o)o(v)o(er)d Fn(k)h Fr(extends)f(the)g(v)m(aluation)g(ring)f Fn(R)1668 127 y Fl(0)1705 120 y Fr(=)e Fn(k)r Fr([)p Fn(z)r Fr(])1839 129 y Fl(\()p Fk(z)q Fl(\))1910 120 y Fr(of)0 190 y Fn(k)r Fr(\()p Fn(z)r Fr(\).)k(W)l(e)g(see)g(that)g Fn(v)425 197 y Fk(R)458 190 y Fr(\()p Fn(z)r Fr(\))f Fn(>)e Fr(0)j(i\013)f Fn(v)719 197 y Fk(R)770 190 y Fr(restricts)g(to)h(the)g(v)m(aluation)f (of)h Fn(k)r Fr(\()p Fn(z)r Fr(\))g(corresp)q(onding)e(to)i(the)0 259 y(prime)d(elemen)o(t)h Fn(z)j Fr(of)e Fn(k)r Fr([)p Fn(z)r Fr(].)f(So)g(for)g(the)h(p)q(ositiv)o(e)f(terms)g(of)h(the)g(ab) q(o)o(v)o(e)f(sum)f(w)o(e)i(ha)o(v)o(e)100 292 y Fh(P)152 344 y Fk(v)172 349 y Fc(R)200 344 y Fl(\()p Fk(z)q Fl(\))p Fk(>)p Fl(0)314 329 y Fn(v)338 336 y Fk(R)371 329 y Fr(\()p Fn(z)r Fr(\)[)p Fn(R=m)p 511 337 44 2 v 13 x Fk(R)602 329 y Fr(:)c Fn(k)r Fr(])h(=)737 292 y Fh(P)790 344 y Fk(Rov)q(er)q(R)929 349 y Fd(0)960 329 y Fn(e)p Fr(\()p Fn(R=R)1103 336 y Fl(0)1125 329 y Fr(\))p Fn(f)5 b Fr(\()p Fn(R=R)1293 336 y Fl(0)1317 329 y Fr(\))14 b(=)g([)p Fn(K)k Fr(:)13 b Fn(k)r Fr(])100 405 y(\(again,)20 b(taking)i(for)e (gran)o(ted)g(equalit)o(y)i(in)e(the)i(ab)q(o)o(v)o(e)e(theorem,)h(in)g (the)g(case)g(of)g(a)g(function)0 475 y(\014eld\).)100 545 y(Here)i(the)h(sum)e(is)g(tak)o(en)i(o)o(v)o(er)e(the)i(distinct)e Fn(R)p Fr('s)h(whic)o(h)g(satisfy)g Fn(v)1451 552 y Fk(R)1484 545 y Fr(\()p Fn(z)r Fr(\))j Fn(>)f Fr(0.)e(W)l(e)g(get)h(the)0 615 y(negativ)o(e)c(terms)f(of)i(the)f(sum)f(b)o(y)h(considering)e(the)
j(in)o(v)o(erse)d(of)j Fn(z)h Fr(and)e(doing)f(the)h(same)g(thing)f(w)o (e)0 684 y(just)e(did.)e(Th)o(us)h(w)o(e)g(see)h(that)g(if)f(w)o(e)h (add)f(up)g(p)q(ositiv)o(e)g(and)g(negativ)o(e)h(terms,)e(w)o(e)i(get)g (zero,)f(and)g(our)0 754 y(assertion)f(is)h(pro)o(v)o(ed.)100 864 y(No)o(w)j(w)o(e)g(c)o(hange)f(direction,)g(and)h(review)g(some)g (in\014nite)f(Galois)h(Theory)l(.)f(W)l(e)i(will)e(con)o(tin)o(ue)0 933 y(with)e(this)h(in)f(the)h(next)g(lecture,)f(but)g(let)h(us)f(no)o (w)g(giv)o(e)g(an)g(elemen)o(tary)g(de\014nition:)100 1043 y Fp(De\014nition:)22 b Fr(Let)f Fn(L)g Fr(:)g Fn(K)k Fr(b)q(e)c(a)g(\014eld)f(extension.)g(W)l(e)i(sa)o(y)e(that)h Fn(L)g Fm(\033)g Fn(K)j Fr(is)d Fp(Galois)h Fr(i\013)f(an)o(y)0 1113 y(\(hence)c(all\))f(of)h(the)g(follo)o(wing)e(equiv)m(alen)o(t)i (conditions)e(holds:)100 1183 y(\(a\))i Fn(L)d Fr(=)280 1145 y Fh(S)322 1198 y Fk(M)374 1183 y Fn(M)5 b Fr(,)17 b(where)f(eac)o(h)h Fn(M)i Fm(\033)14 b Fn(K)20 b Fr(is)c(\014nite)h (Galois)e(\(i.e.)i(separable)e(and)h(normal\).)100 1253 y(\(b\))h Fn(L)c Fm(\033)h Fn(K)20 b Fr(is)d(algebraic,)e(separable,)g (and)h(normal.)100 1322 y(\(c\))h Fn(K)h Fr(=)13 b Fn(L)323 1304 y Fj(G)368 1322 y Fr(for)k(some)e(compact)i(subgroup)d Fm(G)19 b Fr(of)e(Aut\()p Fn(L)p Fr(\).)100 1432 y(In)f(condition)g (\(c\),)h Fn(L)506 1414 y Fj(G)551 1432 y Fr(is,)f(of)g(course,)g(the)h (\014xed)f(\014eld)g(of)h Fm(G)s Fr(:)100 1502 y Fn(L)134 1484 y Fj(G)176 1502 y Fr(=)c Fm(f)p Fn(x)i Fm(2)f Fn(L)f Fr(:)h Fn(\033)r(x)h Fr(=)e Fn(x)i Fm(8)e Fn(\033)j Fm(2)e(G)s(g)p Fr(.)f(W)l(e)i(will)f(talk)g(ab)q(out)g(the)h(top)q(ology)f(of)g(Aut\() p Fn(L)p Fr(\))i(next)e(time.)100 1612 y(As)i(an)g(exercise,)h(pro)o(v) o(e)e(the)i(equiv)m(alence)g(of)f(\(a\),)i(\(b\))f(and)e(\(c\).)0 1742 y Fp(Lecture)20 b(10)c Fr(\(Septem)o(b)q(er)g(20,)g(1995\).)0 1812 y(Scrib)q(e:)g(Ezra)g(Miller)0 1942 y Fp(Reference)p Fr(:)h(W)l(eil's)f Fq(Basic)j(Numb)m(er)e(The)m(ory)h Fr(giv)o(es)e(a)h(full)f(accoun)o(t)g(of)g(in\014nite)g(Galois)g (theory)l(.)100 2051 y(Let)h Fn(L=K)j Fr(b)q(e)d(a)f(\014eld)g (extension.)h(Then)f(the)g(follo)o(wing)g(are)g(equiv)m(alen)o(t:)241 2121 y(1.)h Fn(L)f Fr(is)g(a)h(union)e(of)i(\014nite)f(Galois)g (extensions)g(of)h Fn(K)t Fr(.)241 2191 y(2.)g Fn(L=K)j Fr(is)c(algebraic,)f(separable,)g(and)h(normal.)241 2261 y(3.)h Fn(K)g Fr(=)d Fn(L)443 2243 y Fj(G)488 2261 y Fr(for)i(some)g(compact)g Fm(G)h(2)d Fr(Aut)j Fn(L)p Fr(,)f(where)g Fn(L)1325 2243 y Fj(G)1367 2261 y Fr(=\014xed)g(\014eld) g(of)h Fm(G)s Fr(.)241 2331 y(4.)g Fp(De\014nition:)h Fn(L=K)i Fr(is)c(Galois.)100 2401 y(Notice)21 b(that)h(top)q(ology)f (en)o(ters)f(in)o(to)h(the)g(picture)f(here,)h(sp)q(eci\014cally)f(in)h (the)g(automorphism)0 2470 y(groups)13 b Fn(G)i Fr(in)f(\(3\).)i(W)l(e) f(de\014ne)f(a)g(base)h(for)f(the)h(op)q(en)f(sets)h(of)g(Aut)i Fn(L)e Fr(as)f(follo)o(ws:)g(If)h Fn(E)h Fm(\032)e Fn(L)g Fr(is)h(a)f(\014nite)0 2540 y(subset)h(and)g Fn(f)20 b Fr(:)13 b Fn(E)k Fm(!)d Fn(L)h Fr(is)h(\(in)f(the)i(w)o(ords)d(of)i (Prof.)f(Lenstra\))h(\\an)o(y)f(stupid)g(map",)f(then)i(w)o(e)g (de\014ne)670 2670 y Fn(U)704 2677 y Fk(E)r(;f)786 2670 y Fr(=)e Fn(\033)i Fm(2)e Fr(Aut)j Fn(L)d Fr(:)f Fn(\033)r Fm(j)1150 2677 y Fk(E)1198 2670 y Fr(=)h Fn(f)950 2790 y Fr(21)p eop %%Page: 22 22 22 21 bop 0 50 a Fr(to)18 b(b)q(e)h(an)e(op)q(en)h(set)g(in)g(Aut)f Fn(L)p Fr(.)h(Aut)f Fn(L)h Fr(b)q(ecomes)f(a)h(top)q(ological)g(group)e (\(in)i(fact)h(Hausdor\013)t(\))e(with)0 120 y(comp)q(osition)e(and)h (in)o(v)o(ersion)f(con)o(tin)o(uous.)100 189 y(Let)25 b Fn(L=K)j Fr(b)q(e)d(Galois.)f(Then)g(the)h(group)f(in)g(\(3\))h(is)f (uniquely)g(determined,)g(namely)g Fn(G)j Fr(=)0 274
y(Aut)84 281 y Fk(K)136 274 y Fn(L)170 246 y Fl(def)178 274 y Fr(=)8 b(Gal\()p Fn(L=K)t Fr(\).)13 b(This)g(group)e(is)i(to)h(b) q(e)f(though)o(t)g(of)g(as)g(a)g(top)q(ological)g(group.)f(If)h([)p Fn(L)h Fr(:)f Fn(K)t Fr(])h Fn(<)f Fm(1)p Fr(,)0 344 y(then)j Fn(G)h Fr(is)f(\014nite.)g(In)h(general,)e Fn(G)i Fr(is)f Fp(pro\014nite)p Fr(,)h(i.e.)f(a)h(pro)s(jectiv)o(e)f(limit)g (of)g(\014nite)h(groups:)128 472 y(Gal\()p Fn(L=K)t Fr(\))363 459 y Fm(\030)363 475 y Fr(=)421 472 y(lim)416 502 y Fm( )-8 b(\000)496 510 y Fk(M)t Fj(\022)p Fk(L)599 472 y Fr(Gal\()p Fn(M)s(=K)t Fr(\))14 b(=)904 432 y Fh(\010)941 472 y Fn(\033)i Fr(=)e(\()p Fn(\033)1085 479 y Fk(M)1130 472 y Fr(\))1149 479 y Fk(M)1207 430 y Fh(\014)1207 460 y(\014)1237 472 y Fn(\033)1265 479 y Fk(M)1310 472 y Fm(j)1324 479 y Fk(M)1366 470 y Fi(0)1396 472 y Fr(=)f Fn(\033)1476 479 y Fk(M)1518 470 y Fi(0)1551 472 y Fr(if)k Fn(M)1650 452 y Fj(0)1678 472 y Fm(\022)d Fn(M)1793 432 y Fh(\011)0 627 y Fr(\(where)k(the)h(in)o(v)o(erse)e(limit)h(is)g(o)o (v)o(er)g Fn(M)23 b Fm(\022)17 b Fn(L)i Fr(suc)o(h)e(that)i Fn(M)s(=K)j Fr(is)c(\014nite)h(Galois,)e(so)i(Gal)o(\()p Fn(M)s(=K)t Fr(\))h(is)0 697 y(\014nite)g(for)f(all)h Fn(M)5 b Fr(\).)21 b(Ev)o(ery)e(elemen)o(t)h Fn(\033)i Fr(of)e(Gal\()p Fn(L=K)t Fr(\))g(acts)g(on)g(elemen)o(ts)f(of)h(these)g Fn(M)5 b Fr(,)20 b(so)g Fn(\033)r Fm(j)1848 704 y Fk(M)1911 697 y Fr(=)0 766 y Fn(\033)28 773 y Fk(M)73 766 y Fr(,)h(and)f(if)h Fn(M)27 b Fm(\022)21 b Fn(M)447 748 y Fj(0)462 766 y Fr(,)g(then)g(w)o(e)g(m)o(ust)f(ha)o(v)o(e)g Fn(\033)r Fm(j)984 773 y Fk(M)1026 764 y Fi(0)1063 766 y Fr(=)h Fn(\033)1151 773 y Fk(M)1196 766 y Fm(j)1210 773 y Fk(M)1252 764 y Fi(0)1268 766 y Fr(.)g(The)f(isomorphism)e(ab)q(o)o(v)o(e)j(is)f (as)0 836 y(top)q(ological)15 b(groups,)f(the)i(top)q(ology)f(on)g(the) h(in)o(v)o(erse)e(limit)h(b)q(eing)g(induced)f(b)o(y)21 b(lim)1535 866 y Fm( )-8 b(\000)1615 873 y Fk(M)1659 836 y Fr(Gal\()p Fn(M)s(=K)t Fr(\))14 b Fm(\022)0 888 y Fh(Q)47 941 y Fk(M)100 926 y Fr(Gal)o(\()p Fn(M)s(=K)t Fr(\).)19 b(The)g(pro)q(duct)f(on)g(the)h(righ)o(t)f(is)g(a)h(pro)q (duct)f(of)h(\014nite)f(\(hence)h(compact\))g(spaces,)0 995 y(so)f(is)g(itself)h(compact)g(\(b)o(y)f(T)o(yc)o(hono\013)t('s)f (Theorem\),)h(and)g(it)h(is)f(relativ)o(ely)h(easy)f(to)h(sho)o(w)f (that)h(the)0 1065 y(pro)s(jectiv)o(e)d(limit)g(is)g(closed)g(in)g(the) h(pro)q(duct)f(\(with)h(the)g(pro)q(duct)f(top)q(ology\).)100 1135 y(Here)f(are)f(a)h(couple)g(of)g(en)o(tertaining)f(facts:)h(An)o (y)g(bijectiv)o(e)g(con)o(tin)o(uous)e(map)h(from)g(a)h(compact)0 1205 y(space)c(to)h(a)f(Hausdor\013)f(space)h(is)g(a)h(homeomorphism)o (.)d(\(Pro)q(of:)i(The)g(con)o(tin)o(uous)f(image)g(of)i(a)f(compact)0 1274 y(set)17 b(is)g(compact,)f(so)h(the)g(in)o(v)o(erse)f(map)g(is)g (con)o(tin)o(uous.\))f(One)i(can)g(sho)o(w)e(that)j(a)f(top)q(ological) f(group)0 1344 y(is)g(pro\014nite)f(if)i(and)f(only)g(if)h(it)g(is)f (Hausdor\013,)f(compact,)h(and)g(totally)h(disconnected.)0 1454 y Fp(Examples)p Fr(:)e(1.)22 b(lim)326 1484 y Fm( )-8 b(\000)406 1491 y Fk(n)433 1454 y Fp(Z)p Fn(=p)518 1436 y Fk(n)545 1454 y Fp(Z)14 b Fr(=)g Fp(Z)682 1461 y Fk(p)719 1454 y Fr(=)f Fn(p)p Fr(-adic)j(in)o(tegers.)270 1553 y(2.)22 b(lim)325 1583 y Fm( )-8 b(\000)405 1591 y Fk(n)432 1553 y Fp(Z)p Fn(=n)p Fp(Z)14 b Fr(=)629 1540 y(^)624 1553 y Fp(Z)p Fr(.)270 1646 y(3.)22 b(lim)325 1676 y Fm( )-8 b(\000)405 1684 y Fk(n)432 1646 y Fr(Gal\()p Fp(F)565 1653 y Fk(q)584 1644 y Fc(n)612 1646 y Fn(=)p Fp(F)673 1653 y Fk(q)695 1646 y Fr(\))14 b(=)g(Gal\()895 1634 y(\026)878 1646 y Fp(F)914 1653 y Fk(q)936 1646
y Fn(=)p Fp(F)997 1653 y Fk(q)1019 1646 y Fr(\).)100 1746 y(In)i(fact,)h(Examples)f(2)g(and)g(3)h(are)f(isomorphic,)e(in)i (a)h(canonical)e(w)o(a)o(y)l(.)h(If)h(w)o(e)f(let)h Fp(1)d Fr(=)f(\(1\))1812 1753 y Fk(n)1854 1746 y Fm(2)1906 1733 y Fr(^)1901 1746 y Fp(Z)p Fr(,)0 1816 y(where)23 b(the)g(op)q(eration)g (is)g(addition,)f(then)i Fp(1)f Fr(corresp)q(onds)e(to)j(the)f(F)l(rob) q(enius)f(elemen)o(t)g(F)l(rob)g(=)0 1885 y(\()p Fn(\033)47 1892 y Fk(n)75 1885 y Fr(\))94 1892 y Fk(n)136 1885 y Fm(2)16 b Fr(Gal)o(\()298 1873 y(\026)281 1885 y Fp(F)317 1892 y Fk(q)340 1885 y Fn(=)p Fp(F)401 1892 y Fk(q)423 1885 y Fr(\),)h(where)g(the)g(op)q(eration)g(is)g(comp)q(osition)f(and) g Fn(\033)1383 1892 y Fk(n)1411 1885 y Fr(:)8 b Fp(F)1469 1892 y Fk(q)1488 1883 y Fc(n)1531 1885 y Fm(!)14 b Fp(F)1631 1892 y Fk(q)1650 1883 y Fc(n)1695 1885 y Fr(via)j Fn(x)f Fm(7!)f Fn(x)1914 1867 y Fk(q)1936 1885 y Fr(.)0 1955 y(Note)i(that)g(F)l(rob\()p Fn(x)p Fr(\))d(=)f Fn(x)29 b Fm(\()-8 b(\))27 b Fn(x)666 1937 y Fk(q)702 1955 y Fr(=)13 b Fn(x)29 b Fm(\()-8 b(\))27 b Fn(x)14 b Fm(2)g Fp(F)1055 1962 y Fk(q)1077 1955 y Fr(.)0 2065 y Fp(\\Main)20 b(Theorem)d(of)i(In\014nite)g(Galois)h(Theory")100 2134 y Fr(Let)d Fn(L=K)j Fr(b)q(e)d(Galois)f(with)g(group)g(G.)g(Then)g(the) h(maps)336 2222 y Fh(\010)366 2263 y Fn(E)419 2220 y Fh(\014)419 2250 y(\014)449 2263 y Fn(K)h Fm(\022)c Fn(E)i Fm(\022)e Fn(L)702 2222 y Fh(\011)761 2263 y Fm( )-8 b(!)866 2222 y Fh(\010)895 2263 y Fn(H)19 b Fm(\022)13 b(G)1054 2220 y Fh(\014)1054 2250 y(\014)1084 2263 y Fn(H)21 b Fr(is)16 b(a)h(closed)f(subgroup)1584 2222 y Fh(\011)0 2391 y Fr(\(with)i(the)g(forw)o(ards)d(map)i(de\014ned)g(b) o(y)g Fn(E)h Fm(7!)e Fr(Gal)o(\()p Fn(L=E)s Fr(\))i(and)f(the)h(rev)o (erse)f(map)f(de\014ned)h(b)o(y)g Fn(H)j Fm(7!)0 2461 y Fn(L)34 2443 y Fk(E)68 2461 y Fr(\))g(are)f(inclusion-rev)o(ersi)o (ng)e(in)o(v)o(erse)h(maps.)g(If)i Fn(E)i Fm($)c Fn(H)t Fr(,)i(then)g(\()p Fn(E)s(=K)j Fr(is)c(Galois\))39 b Fm(\()-8 b(\))37 b Fr(\()p Fn(H)25 b Fr(is)0 2531 y(normal)19 b(in)i Fm(G)s Fr(\),)g(and)f(in)h(that)g(case)g(Gal\()p Fn(E)s(=K)t Fr(\))957 2517 y Fm(\030)957 2533 y Fr(=)1017 2531 y Fm(G)s Fn(=H)t Fr(.)g([N.B.)g(Chec)o(k)g(con)o(tin)o(uit)o(y)f (of)h(reduction)0 2600 y(with)f(resp)q(ect)h(to)g(the)f(top)q(ologies)g (in)o(v)o(olv)o(ed.])f(F)l(urthermore,)f(\()p Fn(E)23 b Fr(is)d(\014nite)h(o)o(v)o(er)e Fn(K)t Fr(\))41 b Fm(\()-8 b(\))40 b Fr(\()p Fn(H)26 b Fr(is)0 2670 y(op)q(en)18 b(in)g Fm(G)s Fr(\))35 b Fm(\()-8 b(\))33 b Fr(\()p Fn(H)24 b Fr(is)18 b(closed)g(and)g([)p Fm(G)h Fr(:)e Fn(H)t Fr(])h Fn(<)e Fm(1)p Fr(\).)j(T)l(o)f(pro)o(v)o(e)g(the)h(last)f(equiv) m(alence,)h(w)o(e)f(note)950 2790 y(22)p eop %%Page: 23 23 23 22 bop 0 50 a Fr(that)23 b(\()p Fn(H)k Fr(is)22 b(op)q(en\))h (implies)e(\(i\))i(eac)o(h)f(coset)h(of)g Fn(H)k Fr(is)22 b(op)q(en,)g(so)g(that)h Fn(H)k Fr(is)22 b(the)h(complemen)o(t)e(of)0 120 y(the)f(op)q(en)f(set)293 82 y Fh(S)335 134 y Fk(\033)q(H)s Fj(6)p Fl(=)p Fk(H)471 120 y Fn(\033)r(H)25 b Fr(and)19 b(hence)g Fn(H)24 b Fr(is)19 b(closed;)g(\(ii\))h(the)g(cosets)g(of)g Fn(H)k Fr(constitute)c(an)f(op)q(en)0 193 y(co)o(v)o(er)h(\(whic)o(h)h (is)g(actually)g(a)g(partition\))g(of)g(the)g(compact)g(set)h Fm(G)s Fr(,)e(so)h(that)h(there)f(can)g(b)q(e)g(only)g(a)0 263 y(\014nite)f(n)o(um)o(b)q(er)f(of)h(them)g(\(since)h(an)o(y)f(prop) q(er)f(subset)h(is)g(not)g(a)h(co)o(v)o(er\).)f(The)g(rev)o(erse)f (implication)0 332 y(is)g(pro)o(v)o(ed)g(the)h(same)f(w)o(a)o(y)g (\(using)g(the)h(fact)h(that)f(a)g(\014nite)g(union)e(of)i(closed)f
(sets)h(is)g(closed\).)f(An)o(y)0 402 y(statemen)o(ts)h(ab)q(o)o(v)o(e) h(that)g(ha)o(v)o(e)f(not)h(b)q(een)f(pro)o(v)o(en)g(here)g(can)h(b)q (e)g(tak)o(en)f(as)h(exercises,)f(as)g(can)h(the)0 472 y(follo)o(wing)16 b(standard)f(generalization)h(of)h(\014nite)f(Galois) h(theory:)f(Supp)q(ose)g(that)h(the)g(\014elds)f Fn(E)1801 479 y Fl(1)1838 472 y Fm(\023)e Fn(E)1928 479 y Fl(2)0 542 y Fr(corresp)q(ond)e(to)h Fn(H)349 549 y Fl(1)386 542 y Fm(\022)g Fn(H)479 549 y Fl(2)502 542 y Fr(;)g(then)h([)p Fn(E)691 549 y Fl(1)726 542 y Fr(:)g Fn(E)791 549 y Fl(2)813 542 y Fr(])g Fn(<)f Fm(1)28 b(\()-8 b(\))27 b Fr([)p Fn(H)1145 549 y Fl(2)1181 542 y Fr(:)14 b Fn(H)1250 549 y Fl(1)1273 542 y Fr(])f Fn(<)h Fm(1)p Fr(.)f(W)l(e)h(can)f(try)g(to)h (generalize)0 611 y(in)i(other)g(w)o(a)o(ys,)f(e.g.)h(try)g(the)h (statemen)o(t:)f(Supp)q(ose)f Fn(K)i Fm(\022)d Fn(E)s(;)8 b(F)20 b Fm(\022)14 b Fn(L)i Fr(with)g Fn(E)j Fr(Galois)d(o)o(v)o(er)f Fn(K)t Fr(;)h(then)0 681 y Fn(E)s(=E)6 b Fm(\\)t Fn(F)19 b Fr(and)13 b Fn(E)s(F)t(=F)19 b Fr(are)13 b(Galois,)e(with)i (isomorphic)e(Galois)h(groups)f(\(isomorphism)f(as)i(top)q(ological)0 751 y(groups\).)100 821 y(It)k(turns)f(out)h(that)g(the)g(main)f(in)o (terest)h(is)f(in)h(\014nite)g(algebraic)e(extensions.)i(So)f(wh)o(y)h (should)e(w)o(e)0 890 y(in)o(tro)q(duce)j(in\014nite)g(Galois)h(theory) g(just)g(to)g(do)g(\014nite)f(things?)g(Because)h(it)g(ma)o(y)g(b)q(e)g (necessary)f(\(or)0 960 y(con)o(v)o(enien)o(t\))f(to)g(c)o(ho)q(ose)g (a)h(\014eld)f(b)q(efore)g(y)o(ou)g(start)g(that)h(is)f(large)g(enough) g(to)g(encompass)f(all)h(of)h(the)0 1030 y(\014elds)c(y)o(ou)g(are)h (in)o(terested)f(in,)g(instead)g(of)h(enlarging)f(y)o(our)f(\014nite)i (extension)g(ev)o(ery)g(time)f(it)h(b)q(ecomes)0 1100 y(necessary)l(.)h(Therefore,)h(w)o(e)g(mak)o(e)g(the)h(follo)o(wing)f (de\014nition:)0 1209 y Fp(De\014nition)p Fr(:)i Fn(K)325 1191 y Fk(sep)399 1209 y Fr(=)c Fn(K)494 1216 y Fk(s)529 1209 y Fr(=)g Fm(f)p Fn(\013)g Fm(2)g Fn(K)j Fm(j)d Fn(\013)j Fr(separable)e(o)o(v)o(er)g Fn(K)t Fm(g)p Fr(.)0 1319 y(Note)20 b(that)f(this)g(construction)f(dep)q(ends)g(on)g(\014rst)g(c) o(ho)q(osing)g(an)h(algebraic)f(closure)1664 1306 y(\026)1651 1319 y Fn(K)k Fr(for)d Fn(K)t Fr(.)g(If)g(a)0 1388 y(di\013eren)o(t)d (algebraic)g(closure)g(w)o(ere)g(c)o(hosen,)g(there)h(w)o(ould)f(b)q(e) h(no)g(canonical)f(isomorphism)d(sending)0 1458 y(one)j(separable)f (closure)h(to)h(the)f(other.)0 1568 y Fp(De\014nition)p Fr(:)i Fm(G)309 1575 y Fk(K)361 1568 y Fr(=)13 b(Gal\()p Fn(K)556 1550 y Fk(sep)617 1568 y Fn(=K)t Fr(\))k(is)f(called)g(the)h Fp(absolute)i(Galois)h(group)15 b Fr(of)i Fn(K)t Fr(.)0 1677 y Fp(Remark)p Fr(:)e(Sending)g Fn(K)463 1653 y Fk(sep)459 1691 y Fl(1)538 1677 y Fm(!)e Fn(K)647 1653 y Fk(sep)643 1691 y Fl(2)725 1677 y Fr(results)i(in)h(an)g(inner)g(automorphism)d (of)k Fm(G)1554 1684 y Fk(K)1592 1677 y Fr(.)0 1787 y Fp(T)-5 b(op)r(ological)20 b(Remark)p Fr(:)15 b(If)h Fn(X)21 b Fr(is)16 b(a)g(top)q(ological)g(space)g(with)g(decen)o(t)g (conncectedness)f(prop)q(erties)0 1857 y(\(i.e.)j(lo)q(cally)f (path-connected,)g(path)g(connected,)g(and)g(semilo)q(cally)f (simply-connected\),)g(then)i(the)0 1926 y(category)f(of)h(co)o(v)o (erings)d(of)j Fn(X)j Fr(can)c(b)q(e)g(describ)q(ed)f(b)o(y)h(means)f (of)i(the)f(fundamen)o(tal)e(group.)h(W)l(e)i(will)0 1996 y(go)d(in)o(to)f(this)g(construction)g(\(as)h(w)o(ell)f(as)g(its)h (relations)f(to)h(the)g(description)e(of)i(Galois)f(theory)h(ab)q(o)o (v)o(e\))0 2066 y(next)i(time.)0 2193 y Fp(Lecture)j(11)c Fr(\(Septem)o(b)q(er)g(22,)g(1995\).)0 2263 y(Scrib)q(e:)g(Ebrahim)e (Jahangard)h(Mah)o(b)q(o)q(ob)100 2390 y(Useful)h(references)g(for)g (in\014nite)g(Galois)g(theory:)0 2460 y(1.)g(Artin,)h(Algebraic)e(Num)o (b)q(ers)h(and)f(Algebraic)h(F)l(unctions)f(\(Chapter)h(6,)h(Section)f
(1\).)0 2529 y(2.)g(Cassels-F)l(r\177)-25 b(ohlic)o(h)o(,)14 b(Algebraic)i(Num)o(b)q(er)f(Theory)h(\(Chapter)g(5\).)100 2670 y(W)l(e)g(w)o(an)o(t)g(to)h(reform)o(ulate)e(the)i(main)e(Galois)h (theorem.)g(Analogy)g(with)h(algebraic)e(top)q(ology:)950 2790 y(23)p eop %%Page: 24 24 24 23 bop 100 50 a Fr(Let)21 b Fn(X)j Fr(b)q(e)d(a)f(connected)g(top)q (ological)g(space.)g(Assume)g(that)g(ev)o(ery)h Fn(x)f Fm(2)h Fn(X)j Fr(has)c(a)g(path)o(wise)0 120 y(connected,)h(simply)f (connected)h(op)q(en)g(neigh)o(b)q(orho)q(o)q(d,)e Fn(U)1142 127 y Fk(x)1167 120 y Fr(.)i(A)g(co)o(v)o(ering)f(of)h Fn(X)k Fr(\(at)d Fn(x)1700 127 y Fl(0)1723 120 y Fr(\))f(is)g(a)g(pair) 0 189 y(of)i(a)f(top)q(ological)g(space)g Fn(Y)34 b Fr(and)21 b(a)i(con)o(tin)o(uous)d(map)i Fn(f)29 b Fr(:)24 b Fn(Y)34 b Fm(!)24 b Fn(X)j Fr(suc)o(h)21 b(that)i(there)f(exists)g(an)0 259 y(isomorphism)17 b(b)q(et)o(w)o(een)i Fn(f)519 241 y Fj(\000)p Fl(1)573 259 y Fn(U)607 266 y Fk(x)630 271 y Fd(0)672 259 y Fr(and)g Fn(U)806 266 y Fk(x)829 271 y Fd(0)864 259 y Fm(\002)12 b Fn(E)s Fr(,)20 b(where)f Fn(E)k Fr(is)c(some)g(discrete)g(top)q(ological)h(space,)0 329 y(and)c(so)g(w)o(e)g(ha)o(v)o(e)g(the)h(follo)o(wing)e(comm)o (utativ)o(e)h(diagram:)605 451 y Fn(Y)114 b Fm(\033)102 b Fn(f)918 433 y Fj(\000)p Fl(1)972 451 y Fn(U)1006 458 y Fk(x)1029 463 y Fd(0)1101 438 y Fm(\030)1101 454 y Fr(=)1189 451 y Fn(U)1223 458 y Fk(x)1246 463 y Fd(0)1279 451 y Fm(\002)11 b Fn(E)582 541 y(f)611 469 y Fh(?)611 498 y(?)611 528 y(?)611 558 y(y)939 541 y Fn(f)968 469 y Fh(?)968 498 y(?)968 528 y(?)968 558 y(y)1189 541 y Fm(.)j Fn(\031)599 627 y(X)54 b Fm(3)14 b Fn(x)769 634 y Fl(0)806 627 y Fm(2)133 b Fn(U)1006 634 y Fk(x)1029 639 y Fd(0)0 749 y Fr(Where)16 b Fn(\031)j Fr(is)d(the)h(natural)e(pro) s(jection)h(map.)100 890 y(Let)k Fm(C)218 897 y Fl(1)260 890 y Fr(b)q(e)g(the)g(category)g(whose)g(ob)s(jects)f(are)h(co)o(v)o (erings)e(of)i Fn(X)k Fr(and)19 b(whose)g(morphisms)d(are)0 960 y(de\014ned)g(as)g(follo)o(ws:)0 1030 y(If)j Fn(f)76 1037 y Fl(1)116 1030 y Fr(:)e Fn(Y)176 1037 y Fl(1)216 1030 y Fm(!)g Fn(X)22 b Fr(and)c Fn(f)469 1037 y Fl(2)509 1030 y Fr(:)g Fn(Y)570 1037 y Fl(2)609 1030 y Fm(!)f Fn(X)23 b Fr(are)18 b(t)o(w)o(o)g(co)o(v)o(erings)f(of)i Fn(X)t Fr(,)g(then)f(morphisms)e(b)q(et)o(w)o(een)i(them)0 1100 y(are)e(the)h(con)o(tin)o(uous)d(maps)i(from)f Fn(Y)695 1107 y Fl(1)734 1100 y Fr(to)i Fn(Y)824 1107 y Fl(2)863 1100 y Fr(suc)o(h)e(that)i(the)g(follo)o(wing)e(diagram)g(is)h(comm)o (utativ)o(e:)775 1230 y Fn(Y)804 1237 y Fl(1)965 1200 y Fk(g)935 1230 y Fm(\000)-9 b(!)109 b Fn(Y)1153 1237 y Fl(2)775 1290 y Fn(f)799 1297 y Fl(1)835 1290 y Fm(&)180 b(.)14 b Fn(f)1153 1297 y Fl(2)952 1350 y Fn(X)0 1475 y Fr(No)o(w,)g(for)h Fn(x)228 1482 y Fl(0)264 1475 y Fm(2)f Fn(X)t Fr(,)h(let)g Fn(\031)483 1482 y Fl(1)506 1475 y Fr(\()p Fn(X)q(;)8 b(x)617 1482 y Fl(0)640 1475 y Fr(\))15 b(b)q(e)g(the)g(fundamen)o(tal)e(group)g(of)i Fn(X)k Fr(\(at)c(base)f Fn(x)1579 1482 y Fl(0)1602 1475 y Fr(\).)h(Eac)o(h)e(co)o(v)o(ering)0 1545 y Fn(f)25 b Fr(:)18 b Fn(Y)30 b Fm(!)19 b Fn(X)24 b Fr(\(at)c Fn(x)385 1552 y Fl(0)408 1545 y Fr(\))g(giv)o(es)f(rise)g(to)h(a)f(set,)h (namely)f(the)h(\014b)q(er)f Fn(f)1282 1527 y Fj(\000)p Fl(1)1336 1545 y Fn(x)1364 1552 y Fl(0)1387 1545 y Fr(,)g(pro)o(vided)f (with)i(an)f(action)0 1614 y(of)j Fn(\031)90 1621 y Fl(1)112 1614 y Fr(\()p Fn(X)q(;)8 b(x)223 1621 y Fl(0)247 1614
y Fr(\).)22 b(\(If)g(a)g(set)f(equipp)q(ed)g(with)h(a)f(group)f(action) i Fn(\031)1217 1621 y Fl(1)1240 1614 y Fr(\()p Fn(X)q(;)8 b(x)1351 1621 y Fl(0)1374 1614 y Fr(\))22 b(is)f(giv)o(en,)g(then)g (one)h(could)0 1684 y(uniquely)16 b(determine)g(the)g(co)o(v)o (ering.\))100 1754 y(Let)k Fm(C)218 1761 y Fl(2)260 1754 y Fr(denote)g(the)g(category)f(whose)g(ob)s(jects)h(are)f(sets)h(pro)o (vided)e(with)h(a)h Fn(\031)1624 1761 y Fl(1)1646 1754 y Fr(\()p Fn(X)q(;)8 b(x)1757 1761 y Fl(0)1781 1754 y Fr(\)-action)0 1824 y(and)14 b(whose)f(morphisms)e(are)j(set)h (theoretic)f(maps)f(resp)q(ecting)h(the)g(group)f(action.)h(I.e.)h(if)f Fn(E)1760 1831 y Fl(1)1796 1824 y Fr(and)g Fn(E)1928 1831 y Fl(2)0 1894 y Fr(are)k(t)o(w)o(o)g(sets)g(with)g Fn(\031)423 1901 y Fl(1)446 1894 y Fr(\()p Fn(X)q(;)8 b(x)557 1901 y Fl(0)581 1894 y Fr(\))18 b(acting)g(on)g(them)g(and)g Fn(f)k Fr(:)17 b Fn(E)1182 1901 y Fl(1)1221 1894 y Fm(!)f Fn(E)1324 1901 y Fl(2)1365 1894 y Fr(is)i(a)g(morphism)d(\(a)k(set)g (map\))0 1963 y(then)d Fm(8)p Fn(\033)f Fm(2)f Fn(\031)260 1970 y Fl(1)283 1963 y Fr(\()p Fn(X)q(;)8 b(x)394 1970 y Fl(0)418 1963 y Fr(\))14 b(&)g Fm(8)p Fn(x)f Fm(2)h Fn(E)658 1970 y Fl(1)696 1963 y Fr(w)o(e)j(ha)o(v)o(e)e Fn(f)5 b Fr(\()p Fn(\033)r(x)p Fr(\))17 b(=)c Fn(\033)r(f)5 b Fr(\()p Fn(x)p Fr(\).)100 2104 y Fp(Theorem)p Fr(:)14 b(The)j(ab)q(o)o(v)o(e)f(asso)q(ciation)g(giv)o(es)g(an)g(equiv)m (alence)h(of)f(categories,)g Fn(F)21 b Fr(:)14 b Fm(C)1713 2111 y Fl(1)1749 2104 y Fm(\000)-8 b(!)13 b(C)1869 2111 y Fl(2)1892 2104 y Fr(.)100 2245 y(So,)j(w)o(e)g(ha)o(v)o(e)600 2355 y Fn(H)t(om)p Fr(\()p Fn(Y)761 2362 y Fl(1)784 2355 y Fn(;)8 b(Y)846 2362 y Fl(2)868 2355 y Fr(\))926 2299 y Fk(F)926 2327 y Fj(\030)901 2355 y Fm(\000)-8 b(!)14 b Fn(H)t(om)p Fr(\()p Fn(F)7 b(Y)1196 2362 y Fl(1)1218 2355 y Fn(;)h(F)f(Y)1308 2362 y Fl(2)1331 2355 y Fr(\))0 2461 y(for)16 b(co)o(v)o(erings)f(\()p Fn(Y)339 2468 y Fl(1)362 2461 y Fn(;)8 b(f)408 2468 y Fl(1)431 2461 y Fr(\))17 b(and)e(\()p Fn(Y)611 2468 y Fl(2)634 2461 y Fn(;)8 b(f)680 2468 y Fl(2)703 2461 y Fr(\))17 b(of)g Fn(X)t Fr(.)100 2531 y(Note:)12 b(The)f(degree)g(of)h(a)f(co)o(v)o (ering)f(in)h Fm(C)833 2538 y Fl(1)867 2531 y Fr(\(i.e.)h(n)o(um)o(b)q (er)d(of)j(originals)e(of)h(a)h(p)q(oin)o(t,)e Fn(x)1639 2538 y Fl(0)1662 2531 y Fr(\))i(corresp)q(onds)0 2600 y(to)23 b(cardinalit)o(y)d(of)j(the)f(set)h(with)f Fn(\031)706 2607 y Fl(1)728 2600 y Fr(\()p Fn(X)q(;)8 b(x)839 2607 y Fl(0)863 2600 y Fr(\)-action)22 b(in)g Fm(C)1144 2607 y Fl(2)1167 2600 y Fr(.)g(Also,)g(connectedness)f(of)h(a)g(co)o(v)o (ering)0 2670 y(corresp)q(onds)14 b(to)j(transitivit)o(y)f(of)h(the)g Fn(\031)764 2677 y Fl(1)787 2670 y Fr(\()p Fn(X)q(;)8 b(x)898 2677 y Fl(0)921 2670 y Fr(\)-action.)950 2790 y(24)p eop %%Page: 25 25 25 24 bop 100 50 a Fr(Reform)o(ulation)14 b(of)j(the)g(main)e(theorem)h (of)h(Galois)e(theory:)100 191 y(Let)e Fn(K)j Fr(b)q(e)c(a)g(\014eld)g (and)f(c)o(ho)q(ose)h(a)g(separable)f(closure,)g Fn(K)1167 173 y Fk(sep)1240 191 y Fr(of)h Fn(K)t Fr(.)g(\(Choice)g(of)g(this)g (\014eld)g Fn(K)1837 173 y Fk(sep)1911 191 y Fm(\033)0 261 y Fn(K)18 b Fr(corresp)q(onds)11 b(to)j(a)g(c)o(hoice)f(of)h(an)f (em)o(b)q(edding)f Fm(f)p Fn(x)994 268 y Fl(0)1017 261 y Fm(g)h(\000)-8 b(!)13 b Fn(X)t Fr(.\))h(W)l(rite)g Fm(G)1407 268 y Fk(K)1459 261 y Fr(=)g Fn(Gal)q Fr(\()p Fn(K)1658 243 y Fk(sep)1719 261 y Fn(=K)t Fr(\).)g(Then)0 331 y(there)i(are)h(an)o(ti-equiv)m(alences)e(of)i(categories:)100
472 y Fp(Theorem)p Fr(:)163 551 y Fh(8)163 596 y(>)163 611 y(>)163 626 y(>)163 641 y(>)163 656 y(>)163 671 y(:)225 586 y Fr(\014nite)d(separable)e(\014eld)215 646 y(extensions)h(of)18 b Fn(K)g Fr(with)281 706 y Fn(K)t Fm(\000)p Fr(morphisms)675 551 y Fh(9)675 596 y(>)675 611 y(>)675 626 y(>)675 641 y(>)675 656 y(>)675 671 y(;)761 595 y Fk(F)736 624 y Fm(\000)-9 b(!)761 655 y Fk(G)736 683 y Fm( )h(\000)833 551 y Fh(8)833 596 y(>)833 611 y(>)833 626 y(>)833 641 y(>)833 656 y(>)833 671 y(:)928 586 y Fr(\014nite)13 b(sets)h(pro)o(vided)e(with)i(con)o(tin)o(uous)885 646 y(transitiv)o(e)f Fm(G)1134 653 y Fk(K)1173 646 y Fm(\000)p Fr(action)g(with)h(set)g(theoretic)904 706 y(maps)e(compatible)h(with)h (the)g Fm(G)1507 713 y Fk(K)1545 706 y Fm(\000)p Fr(action)1743 551 y Fh(9)1743 596 y(>)1743 611 y(>)1743 626 y(>)1743 641 y(>)1743 656 y(>)1743 671 y(;)100 831 y Fr(The)19 b(category)h(on)f(the)h(righ)o(t)f(side)f(is)i(called)f(\\Algebraic)g (fundamen)o(tal)f(group)g(of)i Fn(K)t Fr(".)f(Let's)0 900 y(denote)e(this)f(category)g(b)o(y)h Fm(AF)5 b(G)641 907 y Fk(K)696 900 y Fr(and)16 b(the)g(one)h(on)f(the)h(righ)o(t)e (side)h(b)o(y)g Fm(F)5 b(S)t(F)1530 907 y Fk(K)1568 900 y Fr(.)100 971 y(Note:)26 b(Since)e(Galois)h(groups)e(ha)o(v)o(e)i(in)g (general)f(more)g(prop)q(erties)g(than)h(the)h(fundamen)o(tal)0 1041 y(groups,)16 b(our)h(ob)s(jects)h(in)g Fm(AF)5 b(G)611 1048 y Fk(K)668 1041 y Fr(ha)o(v)o(e)17 b(more)g(prop)q(erties:)g (\014niteness)g(of)h(sets)g(and)f(con)o(tin)o(uit)o(y)g(and)0 1110 y(transitivit)o(y)f(of)h(the)g(actions.)100 1252 y(What)f(are)h(the)f(functors)g Fn(F)24 b Fr(and)15 b Fn(G)p Fr(?)597 1383 y Fn(F)21 b Fr(:)14 b Fn(E)i Fm(7\000)-8 b(!)13 b(f)p Fn(K)t Fm(\000)p Fr(maps)26 b(:)14 b Fn(E)i Fm(!)e Fn(K)1267 1363 y Fk(sep)1328 1383 y Fm(g)0 1515 y Fr(And)590 1587 y Fm(fG)645 1594 y Fk(K)683 1587 y Fm(\000)p Fr(maps)26 b(:)13 b Fn(S)k Fm(!)c Fn(K)1047 1567 y Fk(sep)1108 1587 y Fm(g)h( )-8 b(\000)f(7)13 b Fn(S)k Fr(:)c Fn(G:)0 1694 y Fr(Note:)k Fm(fG)189 1701 y Fk(K)227 1694 y Fm(\000)p Fr(maps)26 b(:)14 b Fn(S)i Fm(!)e Fn(K)592 1676 y Fk(sep)652 1694 y Fm(g)j Fr(is)f(a)h(\014eld.) 100 1835 y(One)f(can)g(c)o(hec)o(k)g(that)798 1907 y(id)840 1914 y Fj(F)t(S)s(F)931 1920 y Fc(K)980 1907 y Fm(\000)-9 b(!)14 b Fn(GF)0 2014 y Fr(is)g(an)h(isomorphism)d(of)j(functors.)f (\(If)h(this)g(w)o(ere)f(an)h(equalit)o(y)g(then)f Fn(F)22 b Fr(w)o(ould)14 b(giv)o(e)h(an)f(isomorphism)0 2089 y(of)f(categories)g(instead)f(of)h(equiv)m(alence)g(of)h(them!\))e (I.e.)h Fm(8)p Fn(E)s(;)8 b(E)1182 2055 y Fi(0)1210 2089 y Fm(2)14 b Fr(Ob\()p Fm(F)5 b(S)t(F)1458 2096 y Fk(K)1497 2089 y Fr(\))14 b(&)g Fm(8)p Fn(')e Fm(2)i Fr(Mor)o(\()p Fn(E)s(;)8 b(E)1915 2055 y Fi(0)1931 2089 y Fr(\))0 2158 y(the)17 b(follo)o(wing)e(diagram)g(is)h(comm)o(utativ)o(e:)173 2282 y Fn(E)287 2254 y Fj(\030)262 2282 y Fm(\000)-9 b(!)58 b Fn(GF)7 b Fr(\()p Fn(E)s Fr(\))114 b(=)758 2242 y Fh(\010)787 2282 y Fm(G)817 2289 y Fk(K)855 2282 y Fm(\000)p Fr(maps)27 b(:)13 b Fm(f)p Fn(K)t Fm(\000)p Fr(maps)26 b(:)14 b Fn(E)i Fm(!)e Fn(K)1505 2264 y Fk(sep)1566 2282 y Fm(g)f(!)h Fn(K)1714 2264 y Fk(sep)1775 2242 y Fh(\011)146 2377 y Fn(')179 2305 y Fh(?)179 2335 y(?)179 2365 y(?)179 2394 y(y)437 2305 y(?)437 2335 y(?)437 2365 y(?)437 2394 y(y)470 2377 y Fn(GF)7 b Fr(\()p Fn(')p Fr(\))157 2476 y Fn(E)197 2443 y Fi(0)287 2448 y Fj(\030)262 2476 y Fm(\000)-9 b(!)50 b Fn(GF)7 b Fr(\()p Fn(E)529 2443 y Fi(0)545 2476 y Fr(\))0 2600 y(Little)13 b(exercise:)f(assume)f
(the)h(ab)q(o)o(v)o(e)g(theorem)g(and)f(try)i(to)f(get)h(the)g(main)e (theorem)g(for)h(\\traditional")0 2670 y(\014nite)k(Galois)g(theory)l (.)950 2790 y(25)p eop %%Page: 26 26 26 25 bop 100 50 a Fp(De\014nition)p Fr(:)14 b(Let)f Fn(A)g Fr(b)q(e)g(a)g(ring)f(\(comm)o(utativ)o(e)g(with)g(unit)o(y)g (as)h(usual\))f(and)g Fn(B)j Fr(b)q(e)e(an)f Fn(A)p Fr(-algebra)0 120 y(suc)o(h)h(that)i Fn(B)270 106 y Fm(\030)270 122 y Fr(=)322 120 y Fn(A)359 101 y Fk(n)401 120 y Fr(as)f(an)g Fn(A)p Fr(-mo)q(dule)f(for)h(some)g(p)q(ositiv)o(e)g(in)o(teger)f Fn(n)p Fr(.)i(Then)e Fn(B)k Fr(is)d(called)g(\014nite)g(etale)0 189 y(if)j(for)f(a)g(basis)g Fn(!)317 196 y Fl(1)339 189 y Fn(;)8 b(!)392 196 y Fl(2)415 189 y Fn(;)g(:::;)g(!)532 196 y Fk(n)575 189 y Fr(of)17 b Fn(B)i Fr(o)o(v)o(er)c Fn(A)i Fr(w)o(e)g(ha)o(v)o(e)739 309 y(det)808 269 y Fh(\000)831 309 y Fr(T)l(r)o(\()p Fn(!)932 316 y Fk(i)949 309 y Fn(!)980 316 y Fk(j)1001 309 y Fr(\))1020 269 y Fh(\001)1043 329 y Fk(i;j)1090 309 y Fm(2)d Fn(A)1174 289 y Fj(\003)1197 309 y Fn(:)0 437 y Fr(I.e.)h(the)g(discriminan)n(t)e (is)h(a)h(unit)f(in)g Fn(A)p Fr(.)h(\(In)g(general,)f(w)o(e)g(call)h (something)e(\014nite)i(etale)g(if)g(it)g(is)f(lo)q(cally)0 507 y(lik)o(e)i(the)h(ab)q(o)o(v)o(e.\))100 577 y(If)e Fn(A)f Fr(=)g Fn(K)19 b Fr(is)14 b(a)h(\014eld,)f(then)h(a)g(\014nite)g Fn(K)t Fr(-algebra)e(is)i(\014nite)g(etale)g(if)g(and)g(only)f(if)i(it) f(is)f(isomorphic)0 647 y(as)i Fn(K)t Fr(-algebra)f(to)i(a)g(\014nite)f (pro)q(duct)g(of)g(\014nite)h(separable)e(\014eld)h(extension)g(of)h Fn(K)t Fr(.)100 716 y(No)o(w,)f(if)h(w)o(e)f(cross)g(out)h(the)g(w)o (ords)e(separable)g(\014eld)h(extension)g(of)h Fn(K)k Fr(and)16 b(replace)g(it)h(b)o(y)f(\014nite)0 786 y(etale)k Fn(K)t Fr(-algebra)d(in)i Fm(F)5 b(S)t(F)537 793 y Fk(K)595 786 y Fr(then)19 b(w)o(e)g(can)g(get)g(rid)f(of)i(the)f(transitivit)o (y)g(condition)f(in)h Fm(AF)5 b(G)1851 793 y Fk(K)1908 786 y Fr(in)0 856 y(the)17 b(ab)q(o)o(v)o(e)f(corresp)q(ondence.)100 965 y(Note:)h(if)g Fn(B)h Fr(is)e(\014nite)h(etale)g(then)f Fn(L)11 b Fm(\012)830 972 y Fk(K)879 965 y Fn(B)19 b Fr(is)d(also)g(\014nite)h(etale.)796 1074 y Fn(L)11 b Fm(\012)880 1081 y Fk(K)929 1074 y Fn(B)52 b Fm(\033)h Fn(B)866 1092 y Fh(x)866 1122 y(?)1114 1092 y(x)1114 1122 y(?)866 1194 y Fn(L)119 b Fm(\033)50 b Fn(K)100 1376 y Fr(This)15 b(is)h(ho)o(w)g(Galois)g(theory)h(will)f(lo)q(ok)h (in)f(the)g(21st)h(cen)o(tury!)0 1501 y Fp(Lecture)j(12)c Fr(\(Septem)o(b)q(er)g(25,)g(1995\).)0 1571 y(Scrib)q(e:)g(W)l(a)o(yne) g(Whitney)100 1696 y(Recall)22 b(that)i(last)f(time)g(w)o(e)g (discussed)e(a)i(reform)o(ulation)e(of)i(the)g(fundamen)o(tal)e (theorem)i(of)0 1766 y(Galois)e(Theory:)f(Let)i Fn(K)j Fr(b)q(e)d(a)f(\014eld,)g(and)f(let)i Fn(K)980 1748 y Fk(sep)1062 1766 y Fr(b)q(e)f(a)h(separable)d(closure)i(of)g Fn(K)t Fr(.)g(Let)h Fn(G)1851 1773 y Fk(K)1911 1766 y Fr(=)0 1836 y Fn(Gal)q Fr(\()p Fn(K)146 1818 y Fk(sep)207 1836 y Fn(=K)t Fr(\),)d Fn(G)369 1843 y Fk(K)427 1836 y Fr(is)f(called)g(the)h(absolute)f(Galois)h(group)e(of)i Fn(K)t Fr(.)g(A)g Fn(G)1441 1843 y Fk(K)1480 1836 y Fr(-set)f(is)h (just)f(a)h(set)g(with)0 1905 y(a)g(con)o(tin)o(uous)d(action)j(of)f (the)h(group)f Fn(G)776 1912 y Fk(K)833 1905 y Fr(on)g(it.)h(Then)f(w)o (e)h(ha)o(v)o(e)e(a)i(con)o(tra)o(v)m(arian)o(t)e(equiv)m(alence)i(of)0 1975 y(categories)d(\()p Fm(\003)p Fr(\))293 1982 y Fk(K)332 1975 y Fr(:)258 2095 y Fm(f)p Fr(\014nite)g(separable)f(extensions)h Fn(L)e Fm(\033)g Fn(K)t Fm(g)f($)h(f)p Fr(\014nite)i(transitiv)o(e)g Fn(G)1531 2102 y Fk(K)1570 2095 y Fr(-sets)o Fm(g)0 2215
y Fr(giv)o(en)g(b)o(y:)624 2285 y Fn(L)e Fm(7!)f(f)p Fn(K)t Fr(-em)o(b)q(eddings)h Fn(L)g Fm(\032)g Fn(K)1241 2267 y Fk(sep)1301 2285 y Fm(g)100 2389 y Fr(So)g(no)o(w)g(let)i Fn(R)e Fm(\032)f Fn(K)19 b Fr(b)q(e)c(a)g(discrete)f(v)m(aluation)h (ring)f(and)g(let)1285 2376 y(^)1274 2389 y Fn(R)g Fm(\032)1392 2376 y Fr(^)1379 2389 y Fn(K)19 b Fr(b)q(e)c(the)g(completion)f(of)h Fn(R)p Fr(.)0 2460 y(Cho)q(ose)i(a)g(separable)e(closure)615 2447 y(^)602 2460 y Fn(K)648 2442 y Fk(sep)726 2460 y Fr(of)796 2447 y(^)783 2460 y Fn(K)t Fr(;)i(then)g(the)g(em)o(b)q (edding)e Fn(K)k Fm(\032)1439 2447 y Fr(^)1426 2460 y Fn(K)i Fr(determines)16 b(a)h(sub\014eld)0 2531 y Fn(K)46 2512 y Fk(sep)123 2531 y Fr(of)193 2518 y(^)180 2531 y Fn(K)226 2512 y Fk(sep)303 2531 y Fr(whic)o(h)f(is)g(a)g(separable)f (closure)h(of)g Fn(K)t Fr(.)h(W)l(e)g(ha)o(v)o(e)e(the)i(comm)o(utativ) o(e)f(diagram:)935 2657 y(^)922 2670 y Fn(K)968 2649 y Fk(sep)950 2790 y Fr(26)p eop %%Page: 27 27 27 26 bop 968 50 a Fm(j)899 164 y Fn(K)945 144 y Fk(sep)1019 152 y Fr(^)1005 164 y Fn(K)915 274 y(=)70 b Fm(n)835 388 y Fn(K)881 368 y Fk(sep)1082 376 y Fr(^)1069 388 y Fn(K)915 498 y Fm(n)g Fn(=)871 612 y(K)917 592 y Fk(sep)989 612 y Fm(\\)1046 600 y Fr(^)1033 612 y Fn(K)968 722 y Fm(j)952 832 y Fn(K)100 940 y Fr(Th)o(us)16 b(w)o(e)i(get)g(a)g (restriction)f(map)g(from)g Fn(G)949 947 y Fl(^)939 955 y Fk(K)995 940 y Fr(to)i Fn(G)1097 947 y Fk(K)1135 940 y Fr(.)f(Ev)o(en)o(tually)f(w)o(e)h(will)f(study)h(extensions)0 1018 y(of)71 1005 y(^)57 1018 y Fn(K)j Fr(and)c(shall)f(see)h(that)g Fn(K)570 999 y Fk(sep)644 1005 y Fr(^)631 1018 y Fn(K)h Fr(=)758 1005 y(^)745 1018 y Fn(K)791 999 y Fk(sep)851 1018 y Fr(,)f(so)g(the)g(map)g Fn(G)1191 1025 y Fl(^)1181 1033 y Fk(K)1234 1018 y Fm(!)d Fn(G)1337 1025 y Fk(K)1393 1018 y Fr(is)j(injectiv)o(e.)g(Its)g(image)f(is)h(a)0 1087 y(closed)h(subgroup)d(of)k Fn(G)464 1094 y Fk(K)502 1087 y Fr(,)f(since)g Fn(G)707 1094 y Fl(^)697 1103 y Fk(K)754 1087 y Fr(is)f(compact,)h(whic)o(h)f(is)h(called)f(\\the")i (decomp)q(osition)d(group)0 1157 y Fn(D)h Fr(of)f Fn(R)f Fr(in)g Fn(G)263 1164 y Fk(K)302 1157 y Fr(.)g(\\The")h(b)q(ecause)f (the)g(decomp)q(osition)g(group)f Fn(D)j Fr(is)e(only)g(de\014ned)g(up) f(to)i(conjugacy)0 1229 y(in)g Fn(G)97 1236 y Fk(K)152 1229 y Fr(due)g(to)h(the)g(c)o(hoice)f(of)611 1216 y(^)598 1229 y Fn(K)644 1211 y Fk(sep)704 1229 y Fr(.)0 1362 y Fp(Theorem)e Fr(Let)i Fn(L)e Fm(\033)f Fn(K)20 b Fr(b)q(e)c(a)f (\014nite)g(separable)f(\014eld)h(extension,)g(whic)o(h)g(corresp)q (onds)e(to)j(a)g Fn(G)1835 1369 y Fk(K)1889 1362 y Fr(set)0 1432 y Fn(S)i Fr(under)d(the)h(equiv)m(alence)g(\()p Fm(\003)p Fr(\))600 1439 y Fk(K)639 1432 y Fr(.)f(Then)h(the)g(decomp)q (osition)e(group)g Fn(D)k Fr(acts)e(on)f Fn(S)j Fr(since)e Fn(D)f Fm(\032)f Fn(G)1898 1439 y Fk(K)1936 1432 y Fr(.)0 1502 y(There)i(is)g(a)h(bijection:)219 1634 y Fm(f)p Fr(v)m(aluation)f(rings)f Fn(T)24 b Fr(of)17 b Fn(L)f Fr(extending)h Fn(R)p Fm(g)d($)f Fn(S=D)q Fr(,)k(the)g Fn(D)q Fr(-orbit)f(space)g(of)h Fn(S)0 1771 y Fr(and)g(if)h Fn(T)23 b Fm($)16 b Fn(S)297 1753 y Fj(0)327 1771 y Fm(\032)g Fn(S)s Fr(,)h(then)h Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))p Fn(f)f Fr(\()p Fn(T)h(=R)p Fr(\))18 b(=)e Fm(j)p Fn(S)1007 1753 y Fj(0)1020 1771 y Fm(j)i Fr(and)f(\()p Fm(\003)p Fr(\))1224 1778 y Fl(^)1213 1787 y Fk(K)1270 1771 y Fr(sends)1411 1758 y(^)1405 1771 y Fn(L)1439 1778 y Fk(T)1488 1771 y Fr(to)h Fn(S)1584 1753 y Fj(0)1598 1771 y Fr(.)f(Here)1754
1758 y(^)1749 1771 y Fn(L)1783 1778 y Fk(T)1832 1771 y Fr(refers)0 1841 y(to)g(the)g(completion)e(of)i Fn(L)g Fr(with)f(resp)q(ect)h(to)f(the)h(v)m(aluation)f(de\014ned)g(b)o(y)g Fn(T)7 b Fr(.)0 1974 y Fp(Remark)100 2045 y Fr(Assume)18 b Fn(k)j Fr(=)e Fn(R=m)499 2052 y Fk(R)551 2045 y Fr(is)g(p)q(erfect.)h (Then)f Fn(D)j Fr(has)c(a)i(closed)f(normal)f(subgroup)f Fn(I)t Fr(,)j(the)g(inertia)0 2115 y(group)h(of)h Fn(R)p Fr(,)g(suc)o(h)f(that)h Fn(D)q(=I)631 2101 y Fm(\030)631 2117 y Fr(=)692 2115 y Fn(G)731 2122 y Fk(k)778 2115 y Fr(and)f(for)h(a)g(v)m(aluation)g(ring)f Fn(T)29 b Fr(of)22 b Fn(L)g Fr(extending)g Fn(R)p Fr(,)g(whic)o(h)0 2184 y(corresp)q(onds)14 b(as)j(in)f(the)g(theorem)g(to)h(a)g Fn(G)814 2191 y Fk(K)852 2184 y Fr(-set)g Fn(S)981 2166 y Fj(0)994 2184 y Fr(,)g(then:)100 2255 y(1\))g Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\))14 b(=)g(1)i(\()p Fn(T)24 b Fr(is)16 b(unrami\014ed)e(o)o(v)o(er)i Fn(R)p Fr(\))31 b Fm(\()-8 b(\))30 b Fn(I)20 b Fr(acts)d(trivially)f(on)g Fn(S)1552 2237 y Fj(0)1566 2255 y Fr(.)100 2325 y(2\))h Fn(S)195 2307 y Fj(0)225 2325 y Fr(is)f(the)g(union)g(of)g Fn(f)5 b Fr(\()p Fn(T)h(=R)p Fr(\))18 b Fn(I)t Fr(-orbits)d(of)i Fn(S)s Fr(,)f(eac)o(h)g(of)h(size)f Fn(e)p Fr(\()p Fn(T)6 b(=R)p Fr(\).)100 2396 y(3\))17 b(\()p Fm(\003)p Fr(\))224 2403 y Fk(k)265 2396 y Fr(sends)f Fn(T)6 b(=m)503 2403 y Fk(T)550 2396 y Fr(to)17 b Fn(S)645 2378 y Fj(0)658 2396 y Fn(=I)k Fr(a)16 b Fn(D)q(=I)t Fr(-set.)0 2530 y Fp(De\014nition)21 b(of)d Fn(I)100 2600 y Fr(The)f(v)m(aluation)h (ring)536 2588 y(^)525 2600 y Fn(R)g Fr(of)653 2588 y(^)639 2600 y Fn(K)k Fr(extends)c(uniquely)f(to)h(a)g(v)m(aluation)f(ring)g Fn(U)23 b Fr(of)1644 2588 y(^)1631 2600 y Fn(K)1677 2582 y Fk(sep)1738 2600 y Fr(.)17 b(F)l(urther-)0 2670 y(more,)d Fn(U)s(=m)243 2677 y Fk(U)291 2670 y Fr(is)g(a)h(separable)f(closure)g (of)h Fn(k)r Fr(.)f(The)h(absolute)f(Galois)g(group)g Fn(G)1509 2677 y Fl(^)1499 2686 y Fk(K)1553 2670 y Fr(\014xes)h Fn(U)20 b Fr(and)14 b Fn(m)1857 2677 y Fk(U)1905 2670 y Fr(as)950 2790 y(27)p eop %%Page: 28 28 28 27 bop 0 50 a Fr(sets.)16 b(So)g Fn(G)229 57 y Fl(^)219 66 y Fk(K)273 50 y Fr(acts)h(as)e(\014eld)h(automorphisms)d(on)j Fn(k)990 32 y Fk(sep)1064 50 y Fr(=)d Fn(U)s(=m)1222 57 y Fk(U)1255 50 y Fr(,)j(\014xing)f Fn(k)h Fr(=)d Fn(R=m)1622 57 y Fk(R)1655 50 y Fr(.)j(Th)o(us)f(w)o(e)h(get)0 120 y(a)h(map)e Fn(\036)f Fr(:)f Fn(G)273 127 y Fl(^)262 135 y Fk(K)315 120 y Fm(!)g Fn(G)417 127 y Fk(k)442 120 y Fr(.)k(That)f(is,)g(w)o(e)g(ha)o(v)o(e)g(the)h(follo)o(wing)e (diagram:)822 184 y(^)808 197 y Fn(K)854 179 y Fk(sep)929 197 y Fm(\033)e Fn(U)20 b Fm(!)13 b Fn(k)1126 179 y Fk(sep)766 267 y Fn(G)815 274 y Fl(^)805 283 y Fk(K)860 267 y Fm(j)113 b(j)99 b(j)17 b Fn(G)1170 274 y Fk(k)855 332 y Fr(^)841 344 y Fn(K)35 b Fm(\033)981 332 y Fr(^)970 344 y Fn(R)14 b Fm(!)g Fn(k)0 414 y Fr(One)j(pro)o(v)o(es)f(that)h(the)h(map)f Fn(\036)g Fr(is)f(surjectiv)o(e,)h(and)g(then)g(de\014nes)g Fn(I)h Fr(=)d(k)o(er)8 b Fn(\036)p Fr(.)17 b(If)h(c)o(har)o Fn(k)e Fr(=)f(0,)i Fn(I)1847 401 y Fm(\030)1847 417 y Fr(=)1906 401 y(^)1901 414 y Fp(Z)p Fr(;)0 486 y(if)g Fn(k)h Fr(is)e(\014nite,)g Fn(G)320 493 y Fk(k)359 472 y Fm(\030)359 488 y Fr(=)416 473 y(^)411 486 y Fp(Z)p Fr(.)h(These)f(topics)g(are)g(the)h(b)q(eginning)e(of)i(the)g(study)f (of)h(lo)q(cal)f(\014elds.)0 613 y Fp(Bac)n(k)k(to)f(Curv)n(es)100 683 y Fr(So)e(no)o(w)g(w)o(e)h(consider)e(a)i(\014eld)f
Fn(k)i Fr(and)e(a)h(\014eld)f(extension)h Fn(K)i Fm(\033)15 b Fn(k)20 b Fr(whic)o(h)d(is)g(\014nitely)h(generated)0 753 y(of)h(transcendence)e(degree)h(1.)g(Suc)o(h)f(a)h(\014eld)g (extension)g(w)o(e)g(will)g(call)g(a)g(\\function)g(\014eld)g(of)g(a)h (curv)o(e)0 822 y(o)o(v)o(er)d Fn(k)r Fr(".)100 892 y(W)l(e)j(then)h (de\014ne)f(the)h(\\abstract)f(curv)o(e)g(asso)q(ciated)g(to)h Fn(K)j Fm(\033)c Fn(k)r Fr(",)g(denoted)g(b)o(y)g Fn(C)k Fr(or)c Fn(C)1834 899 y Fk(K)1872 892 y Fr(,)g(as)0 962 y Fn(C)f Fr(=)d Fm(f)p Fr(v)m(aluation)i(rings)f(of)i Fn(K)h Fm(\033)c Fn(k)r Fm(g)p Fr(.)h(W)l(e)i(think)f(of)h(the)f (elemen)o(ts)g(of)g Fn(C)k Fr(as)c(p)q(oin)o(ts,)f(denoted)h(b)o(y)g Fn(P)7 b Fr(.)0 1032 y(If)15 b(w)o(e)g(are)g(faced)g(with)h(the)f (realit)o(y)g(that)g Fn(P)22 b Fr(is)15 b(a)g(ring,)f(then)h(w)o(e)g (usually)f(denote)h(it)h(b)o(y)f Fn(O)1711 1039 y Fk(P)1759 1032 y Fr(\(or)g Fn(O)1876 1039 y Fk(C;P)0 1101 y Fr(or)h Fn(O)99 1108 y Fk(K;P)178 1101 y Fr(\).)100 1171 y Fn(C)i Fr(con)o(tains)d(the)h(\\generic)e(p)q(oin)o(t")h Fn(\030)h Fr(=)e Fn(K)19 b Fr(with)d Fn(O)1083 1178 y Fk(\030)1118 1171 y Fr(=)e Fn(K)t Fr(,)h(and)g(all)g(the)h(other)f(p)q(oin)o(ts)g (of)h Fn(C)i Fr(are)0 1241 y(called)i(\\closed)f(p)q(oin)o(ts".)g(W)l (e)h(kno)o(w)g(that)g(for)g Fn(P)27 b Fr(closed,)20 b Fn(O)1188 1248 y Fk(P)1241 1241 y Fr(is)g(a)g(discrete)f(v)m(aluation)h (ring)f(with)0 1311 y([)p Fn(O)52 1318 y Fk(P)85 1311 y Fn(=m)154 1318 y Fk(P)201 1311 y Fr(:)13 b Fn(k)r Fr(])h(=)f(deg)c Fn(P)21 b(<)13 b Fm(1)0 1438 y Fp(Connection)20 b(with)h(Sc)n(hemes)d (\(Informal\))100 1508 y Fr(W)l(e)i(de\014ne)f(a)h(top)q(ology)g(on)g Fn(C)t Fr(:)f(let)h Fn(X)k Fm(\032)19 b Fn(C)k Fr(b)q(e)e(closed)39 b Fm(\()-8 b(\))38 b Fn(X)24 b Fr(is)c(\014nite)f(with)h Fn(\030)i Fm(62)e Fn(X)t Fr(,)g(or)0 1578 y Fn(X)e Fr(=)c Fn(C)t Fr(.)f(Then)i(if)f Fn(U)20 b Fm(\032)13 b Fn(C)18 b Fr(is)c(op)q(en,)h(let)g Fn(O)q Fr(\()p Fn(U)5 b Fr(\))15 b(=)f Fm(\\)979 1585 y Fk(P)5 b Fj(2)p Fk(U)1070 1578 y Fn(O)1108 1585 y Fk(P)1141 1578 y Fr(.)15 b Fn(O)q Fr(\()p Fn(U)5 b Fr(\))15 b Fm(\032)f Fn(K)t Fr(,)g(and)g Fn(O)i Fr(is)f(a)f(sheaf)h(of)f(rings)0 1648 y(on)i Fn(C)t Fr(,)g(so)g(\()p Fn(C)q(;)8 b(O)q Fr(\))17 b(is)f(a)h(ringed)e(space.) 100 1717 y(In)h(fact)i(\()p Fn(C)q(;)8 b(O)q Fr(\))17 b(is)g(a)g(sc)o(heme.)e(T)l(o)i(b)q(e)g(precise,)f(\()p Fn(C)q(;)8 b(O)q Fr(\))18 b(is)e(an)g(in)o(tegral,)g(1-dimensional)e (sc)o(heme)0 1787 y(o)o(v)o(er)20 b Fn(k)i Fr(whic)o(h)e(is)g(prop)q (er)f(and)h(regular.)g(Regular)f(means)h(the)g(lo)q(cal)h(rings)f Fn(O)1535 1794 y Fk(P)1588 1787 y Fr(are)h(regular)e(lo)q(cal)0 1857 y(rings;)g(prop)q(er)h(is)g(a)h(prop)q(ert)o(y)f(of)h(sc)o(hemes)e (corresp)q(onding)g(to)i(the)g(prop)q(ert)o(y)f(of)h(compactness)e(in)0 1926 y(top)q(ology)l(.)100 1996 y(F)l(urthermore,)d(the)k(category)f (of)h(suc)o(h)e(sc)o(hemes)f(o)o(v)o(er)i Fn(k)i Fr(is)d(an)o(ti-equiv) m(alen)o(t)h(to)g(the)h(category)0 2066 y(of)f(functions)g(\014elds)f (of)h(curv)o(es.)f(This)h(is)f(the)i(reason)e(that)h(w)o(e)g(approac)o (h)f(the)h(study)g(of)g(curv)o(es)f(b)o(y)0 2136 y(considering)d(their) h(functions)f(\014elds,)h(and)g(thereb)o(y)g(a)o(v)o(oid)f(most)h (algebraic)g(geometric)g(detail.)0 2263 y Fp(Lecture)k(13)c Fr(\(Septem)o(b)q(er)g(27,)g(1995\).)0 2333 y(Scrib)q(e:)g(La)o(wren)f (Smitline)0 2461 y(Let)21 b Fn(K)j Fm(\033)c Fn(k)i Fr(b)q(e)f(a)g (\014nitely)f(generated)g(extension)g(of)h(transcendence)e(degree)h(1.) g(W)l(e)h(de\014ne)f Fn(C)j Fr(=)0 2531 y Fn(C)36 2538 y Fk(K)88 2531 y Fr(=)13 b Fm(f)p Fr(v)m(aluation)j(rings)e(of)i Fn(K)j Fr(o)o(v)o(er)c Fn(k)r Fm(g)e(3)h Fn(p)g Fr(=)g Fm(O)971 2538 y Fk(p)993 2531 y Fr(.)i(Note)g(that)g Fn(C)j Fr(is)c(an)h(in)o(tegral)e(one)h(dimensional)0
2600 y(prop)q(er)23 b(regular)f(sc)o(heme)h(o)o(v)o(er)g Fn(k)r Fr(.)h(Prop)q(er)f(means)g(separated,)f(of)i(\014nite)g(t)o(yp)q (e,)g(and)g(univ)o(ersally)0 2670 y(closed.)950 2790 y(28)p eop %%Page: 29 29 29 28 bop 100 50 a Fr(If)17 b(y)o(our)g Fn(C)j Fr(satis\014es)d(these)g (prop)q(erties,)f(in)h(fact,)h Fn(C)j Fr(is)c(a)g(curv)o(e.)g(If)h(y)o (ou)f(lo)q(ok)h(at)f(more)g(general)0 120 y(curv)o(es,)12 b(there)h(ma)o(y)g(b)q(e)h(more)e(than)h(one)g(with)g(the)h(same)f (function)g(\014eld)f Fn(K)t Fr(.)h(The)g(conditions)g(regular)0 189 y(and)j(univ)o(ersally)f(closed)h(corresp)q(ond)e(to)j(nonsingular) d(and)i(pro)s(jectiv)o(e.)100 261 y(T)l(o)e(get)h(curv)o(es)f(without)g (the)h(univ)o(ersal)e(closure)g(prop)q(ert)o(y)l(,)g(include)h(as)g (new)h(curv)o(es)e(nonempt)o(y)0 330 y(op)q(en)h(subsets)f(of)h Fn(C)t Fr(.)f(T)l(o)g(get)i(curv)o(es)e(without)g(the)i(regularit)o(y)d (prop)q(ert)o(y)l(,)h(lo)q(ok)h(at)g(sets)g Fn(C)1709 312 y Fj(0)1736 330 y Fr(consisting)0 400 y(of)24 b(lo)q(cal)g(rings)e (b)q(et)o(w)o(een)i Fn(k)h Fr(and)f Fn(K)j Fr(with)d(\014eld)f(of)h (fractions)f Fn(K)28 b Fr(suc)o(h)23 b(that)h Fn(C)1599 382 y Fj(0)1628 400 y Fm(\\)16 b Fn(C)1713 407 y Fk(K)1775 400 y Fr(is)23 b(op)q(en,)0 470 y(nonempt)o(y)15 b(in)i Fn(C)327 477 y Fk(K)381 470 y Fr(and)f(suc)o(h)f(that)i(an)o(y)f(t)o(w) o(o)h(rings)e(in)h Fn(C)1107 452 y Fj(0)1137 470 y Fr(generate)g Fn(K)k Fr(as)d(a)f(ring.)100 541 y(T)l(o)e(desingularize,)f(replace)i (all)f(reduced)g(fat)i(lo)q(cal)f(rings)f(in)g Fn(C)1307 523 y Fj(0)1336 541 y Fr(b)o(y)g(the)i(v)m(aluation)e(rings)g(whic)o(h) 0 611 y(dominate)i(them.)100 682 y(The)c(term)f(regular)g(means)g(that) i(all)f Fm(O)834 689 y Fk(p)869 682 y Fr(are)g(regular.)e(The)i(term)g (nonsingular)e(means)h(all)h(\012)1853 691 y Fj(O)1885 696 y Fc(p)1905 691 y Fk(=k)0 752 y Fr(are)k(free)h(of)g(rank)e(1)i(o)o (v)o(er)f Fm(O)539 759 y Fk(p)561 752 y Fr(.)h(If)g Fn(k)h Fr(is)e(p)q(erfect,)h(regular)e(implies)g(nonsingular.)0 863 y Fq(Example:)i Fr(Hyp)q(erelliptic)g(curv)o(es.)100 934 y(Let)h Fn(g)g Fm(\025)e Fr(0)p Fn(;)8 b(g)18 b Fm(2)f Fp(Z)p Fr(.)h(Let)h Fn(a;)8 b(b)17 b Fm(2)g Fn(k)r Fr([)p Fn(X)t Fr(])g(suc)o(h)g(that)h(deg)9 b Fn(a)17 b Fm(\024)f Fn(g)r(;)26 b Fr(deg)8 b Fn(b)17 b Fr(=)f(2)p Fn(g)e Fr(+)e(1.)18 b(Let)g(the)h(curv)o(e)0 1004 y(b)q(e)e Fn(Y)108 986 y Fl(2)141 1004 y Fr(+)11 b Fn(a)p Fr(\()p Fn(X)t Fr(\))p Fn(Y)23 b Fr(+)11 b Fn(b)p Fr(\()p Fn(X)t Fr(\))k(=)f(0.)i(Consider)f(the)i(ring)f Fn(k)r Fr([)p Fn(X)q(;)8 b(Y)i Fr(])p Fn(=)p Fr(\()p Fn(Y)1269 986 y Fl(2)1303 1004 y Fr(+)h Fn(aY)22 b Fr(+)11 b Fn(b)p Fr(\))j(=)g Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])p Fn(:)100 1075 y Fr(W)l(e)18 b(sho)o(w)g Fn(Y)353 1057 y Fl(2)388 1075 y Fr(+)12 b Fn(aY)23 b Fr(+)13 b Fn(b)19 b Fr(is)f(irreducible)e (o)o(v)o(er)i Fn(k)r Fr(\()p Fn(X)t Fr(\).)h(If)g(c)o(har)e Fn(k)i Fm(6)p Fr(=)e(2)p Fn(;)h Fr(irreducibilit)o(y)f(is)h(clear,)0 1145 y(since)e Fn(k)r Fr([)p Fn(X)t Fr(])f(is)h(in)o(tegrally)f (closed.)g(If)h(there)g(are)g(ro)q(ots,)g(they)g(ha)o(v)o(e)g(degrees)f (summing)f(to)i(2)p Fn(g)c Fr(+)e(1,)15 b(so)0 1215 y(one)h(ro)q(ot)h (has)f(degree)g(at)h(least)f Fn(g)d Fr(+)e(1,)16 b(so)g(deg)9 b Fn(a)14 b Fm(\025)g Fn(g)e Fr(+)f(1)p Fn(:)j Fm(\)\()100 1286 y Fr(If)i Fn(g)g Fr(=)d(1,)k(w)o(e)f(ha)o(v)o(e)g(an)g(elliptic)g (curv)o(e.)g(F)l(or)g Fn(g)f(>)f Fr(1,)i Fm(1)g Fr(is)g(a)h(singular)e (p)q(oin)o(t.)100 1357 y(So)h Fn(K)i Fr(=)13 b Fn(Q)p Fr(\()p Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(]\))18 b(is)e(a)g(\014eld.)100 1429 y(The)h(equation)h(de\014nes)f(an)h(a\016ne)f(curv)o(e)g(Sp)q(ec)h Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(],)18 b(whic)o(h)f(corresp)q(onds)f(to)i
(the)g(set)g Fn(C)1882 1410 y Fj(0)1911 1429 y Fr(=)0 1498 y Fm(f)p Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])157 1505 y Fe(p)193 1498 y Fr(:)16 b Ff(p)f Fm(2)j Fr(Sp)q(ec)g Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])p Fm(g)p Fr(.)18 b(This)f(a\016ne)g (curv)o(e)g(is)h(regular)e(if)i(and)f(only)h(if)f Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])18 b(is)g(Dedekind,)0 1568 y(or)e Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])17 b(is)g(the)g(in)o(tegral)e (closure)h(of)h Fn(k)r Fr([)p Fn(x)p Fr(])g(in)f Fn(K)t Fr(.)h(If)g(c)o(har)f Fn(k)g Fm(6)p Fr(=)e(2,)i(this)h(holds)e(i\013)i Fn(a)1612 1550 y Fl(2)1646 1568 y Fm(\000)11 b Fr(4)p Fn(b)17 b Fr(is)g(square)0 1638 y(free)g(in)f Fn(k)r Fr([)p Fn(x)p Fr(].)g(It)h(is)f(nonsingular)e(if)i(\001)e(=)f Fn(a)820 1620 y Fl(2)854 1638 y Fm(\000)e Fr(4)p Fn(b)j Fm(6)p Fr(=)g(0.)100 1711 y(If)i Fn(k)h Fr(is)f(not)g(p)q(erfect,)g (then)g(there)g(is)g Fn(a)e Fm(2)g Fn(k)k Fr(suc)o(h)c(that)j Fn(a)f Fr(is)g(not)g Fn(\013)1361 1692 y Fk(p)1400 1711 y Fr(for)f Fn(\013)f Fm(2)g Fn(k)k Fr(and)d(c)o(har)g Fn(k)h Fr(=)d Fn(p)p Fr(.)0 1780 y(So)j Fn(x)97 1762 y Fk(p)132 1780 y Fm(\000)11 b Fn(a)16 b Fr(is)h(irreducible)d(and)i (square)g(free;)g(e.g.)h Fn(y)1002 1762 y Fl(2)1038 1780 y Fr(=)d Fn(x)1119 1762 y Fk(p)1153 1780 y Fm(\000)d Fn(a)17 b Fr(has)f(gen)o(us)1478 1758 y Fk(p)p Fj(\000)p Fl(1)p 1478 1769 72 2 v 1504 1797 a(2)1555 1780 y Fr(.)100 1855 y(If)g(c)o(har)g Fn(k)f Fr(=)f Fn(p)p Fr(,)i Fn(C)k Fr(is)c(nonsingular)e(i\013)j(gcd)o(\()p Fn(a;)8 b Fr(\()p Fn(a)1029 1836 y Fl(2)1053 1855 y Fr(\))1072 1836 y Fj(0)1086 1855 y Fn(b)k Fm(\000)f Fr(\()p Fn(b)1209 1836 y Fj(0)1223 1855 y Fr(\))1242 1836 y Fl(2)1265 1855 y Fr(\))j(=)g(1.)100 1966 y(What)f(are)g(morphisms)d(b)q(et)o(w)o(een)j(these)g(\(actual\))h (curv)o(es?)e(Supp)q(ose)g Fn(L=k)i Fr(and)f Fn(D)i Fr(=)f Fn(C)1766 1973 y Fk(L)1795 1966 y Fr(,)g(where)0 2035 y Fn(L)i Fr(is)f(a)h(\014nitely)g(generated)g(\014eld)f(extension)h(of) g(transcendence)f(degree)g(1.)h(T)l(ranslating)e(the)i(sc)o(heme)0 2105 y(theoretic)j(notion)f(of)g(a)h(morphism)d Fn(D)j Fm(!)e Fn(C)22 b Fr(to)c(\014elds,)g(w)o(e)g(\014nd)g(Hom)1376 2112 y Fl(sc)o(hemes)1517 2105 y Fr(\()p Fn(D)q(;)8 b(C)t Fr(\))19 b(is)f(equiv)m(alen)o(t)0 2175 y(to)g(Hom)166 2182 y Fl(\014elds)e(o)o(v)o(er)g Fk(k)385 2175 y Fr(\()p Fn(K)q(;)8 b(L)p Fr(\),)19 b(where)f Fn(K)i Fm(\032)c Fn(L)p Fr(.)i(In)g(fact,)h(it)f(is)1165 2138 y Fh(S)1206 2175 y Fm(f)p Fr(pairs)f(consisting)g(of)h(a)g(closed)g(p)q(oin)o(t)0 2245 y Fn(P)j Fm(2)14 b Fn(C)20 b Fr(plus)15 b(a)i Fn(k)r Fr(-algebra)d(em)o(b)q(edding)h Fn(k)g Fm(\032)f Fn(k)r Fr(\()p Fn(P)7 b Fr(\))14 b Fm(\032)f Fn(L)p Fm(g)p Fn(:)100 2317 y(C)t Fr(\()p Fn(k)r Fr(\))g(=)g Fm(f)p Fn(P)21 b Fm(2)14 b Fn(C)j Fr(:)d Fn(k)r Fr(\()p Fn(P)7 b Fr(\))14 b(=)f Fn(k)r Fm(g)p Fr(.)100 2390 y(Next)k(time:)g(line)f(bundles)f (and)h(adeles.)0 2462 y Fp(Lecture)k(14.)c Fr(\(Septem)o(b)q(er)g(29,)g (1995\).)0 2533 y(Scrib)q(e:)g(Dylan)g(Th)o(urston)0 2670 y Fp(Coming)i(next:)i(The)f(Riemann-Ro)r(c)n(h)f(Theorem!)950 2790 y Fr(29)p eop %%Page: 30 30 30 29 bop 0 50 a Fr(But)17 b(\014rst,)f(some)f(notation)i(from)e(our)h (sp)q(onsor.)100 120 y(Recall)23 b(that)i(for)e(an)o(y)h(base)f (\014eld)g Fn(k)j Fr(con)o(tained)d(in)g(a)h(function)g(\014eld)f Fn(K)t Fr(,)h(w)o(e)g(ha)o(v)o(e)f(de\014ned)0 189 y(the)e(curv)o(e)f Fn(C)t Fr(,)g(whose)g(p)q(oin)o(ts)g Fn(P)28 b Fr(are)20 b(the)h(v)m(aluation)f(rings)g(of)h Fn(K)j Fr(lying)d(o)o(v)o(er)f Fn(k)r Fr(.)g(As)h(a)f(matter)h(of)0 259 y(notation,)15
b(the)h(ring)f(corresp)q(onding)f(to)i(a)g(p)q(oin)o(t)f Fn(P)23 b Fr(is)15 b(denoted)h Fm(O)1281 266 y Fk(P)1313 259 y Fr(,)g(with)g(maximal)f(ideal)g Ff(m)1815 266 y Fk(P)1848 259 y Fr(.)h(\(So)0 329 y(in)h(some)f(sense)g Fm(O)352 336 y Fk(P)400 329 y Fr(=)f Fn(P)7 b Fr(.\))17 b(W)l(e)g(are)g(usually)f(in)o(terested)g(in)h(the)g(closed)g(p)q(oin)o (ts,)f(whic)o(h)g(corresp)q(ond)0 399 y(to)f(discrete)f(v)m(aluation)g (rings)f(\(that)j(is,)e(all)g(v)m(aluation)g(rings)f(b)q(esides)h(the)h (trivial)f(one)g(with)h Fm(O)1805 406 y Fk(P)1852 399 y Fr(=)e Fn(K)0 468 y Fr(and)j Ff(m)135 475 y Fk(P)182 468 y Fr(=)e(\(0\)\).)100 538 y(The)22 b(divisor)g(class)g(\014eld)g Fm(O)654 545 y Fk(P)687 538 y Fn(=)p Ff(m)750 545 y Fk(P)806 538 y Fr(will)g(b)q(e)h(denoted)g Fn(K)t Fr(\()p Fn(P)7 b Fr(\).)23 b(The)f(degree)h(of)g Fn(k)r Fr(\()p Fn(P)7 b Fr(\))23 b(o)o(v)o(er)f Fn(k)r Fr(,)0 608 y([)p Fn(k)r Fr(\()p Fn(P)7 b Fr(\))23 b(:)f Fn(k)r Fr(],)f(is)h(called)f(the)h (degree)g(of)g Fn(P)7 b Fr(,)21 b(deg)9 b Fn(P)e Fr(;)22 b(it)g(is)f(a)h(p)q(ositiv)o(e)g(in)o(teger.)f Fn(C)t Fr(\()p Fn(K)t Fr(\))g(is)h(the)g(set)g(of)0 677 y(\\rational)15 b(p)q(oin)o(ts")h(of)h Fn(C)t Fr(;)e(it)i(is)f(the)h(set)517 802 y Fn(C)t Fr(\()p Fn(K)t Fr(\))c(=)h Fm(f)p Fn(P)7 b Fm(j)p Fn(P)23 b Fr(a)17 b(closed)e(p)q(oin)o(t,)25 b(deg)8 b Fn(P)21 b Fr(=)14 b(1)p Fm(g)p Fn(:)0 927 y Fr(The)j(v)m(aluation)g Fn(K)i Fm(\))c Fp(Z)d Fm([)g(f1g)17 b Fr(corresp)q(onding)e(to)i Fn(P)25 b Fr(is)17 b(denoted)g(ord)1407 934 y Fk(P)1457 927 y Fr(\(or,)g(o)q(ccassionally)l(,)f Fn(v)1883 934 y Fk(P)1917 927 y Fr(\).)0 1001 y(The)h(completion)g(of)h Fn(K)j Fr(with)c(resp)q(ect)h(to)g Fn(P)7 b Fr(,)17 b(whic)o(h)g(w)o (as)f(formerly)1352 988 y(^)1339 1001 y Fn(K)21 b Fr(\(in)d(an)f(abuse) f(of)i(notation\))0 1070 y(will)e(no)o(w)g(b)q(e)h(denoted)f Fn(K)496 1077 y Fk(P)529 1070 y Fr(.)h(Note)g(that)g(this)f(is)g Fq(not)h Fr(a)f(lo)q(calization.)100 1140 y(Supp)q(ose)i Fn(L=K)23 b Fr(is)d(a)f(\014nite)h(extension.)f(Then)g(there)h(is)f(a)g (natural)g(map)g(b)q(et)o(w)o(een)g(the)h(corre-)0 1210 y(sp)q(onding)c(curv)o(es)h Fn(D)i Fr(and)e Fn(C)t Fr(,)g(de\014ned)f (b)o(y)i Fn(Q)e Fm(2)g Fn(D)h Fm(7!)e Fn(P)23 b Fm(2)16 b Fn(C)21 b Fr(if)c Fm(O)1313 1217 y Fk(Q)1359 1210 y Fm(\\)12 b Fn(K)19 b Fr(=)d Fm(O)1560 1217 y Fk(P)1592 1210 y Fr(.)i(Note)h(that)f(eac)o(h)0 1280 y(p)q(oin)o(t)h(in)g Fn(D)i Fr(maps)d(to)i(a)f(unique)g(p)q(oin)o(t)g(in)g Fn(C)t Fr(,)g(but)g(there)g(ma)o(y)g(b)q(e)h(sev)o(eral)e(p)q(oin)o(ts) h(mapping)e(to)j(a)0 1349 y(giv)o(en)c(p)q(oin)o(t)g(in)f Fn(C)t Fr(.)h(Eac)o(h)f(in)o(v)o(erse)g(image)h(of)g(a)h(p)q(oin)o(t)e Fn(P)21 b Fm(2)14 b Fn(C)20 b Fr(corresp)q(onds)14 b(to)i(an)g (extension)g(of)h(the)0 1419 y(v)m(aluation)e(ring)g Fm(O)358 1426 y Fk(P)391 1419 y Fr(,)h(so)f(w)o(e)h(can)f(use)h(the)g (n)o(um)o(b)q(ers)d(de\014ned)i(earlier:)f Fn(e)p Fr(\()p Fm(O)1433 1426 y Fk(Q)1467 1419 y Fn(=)p Fm(O)1532 1426 y Fk(P)1565 1419 y Fr(\))i(and)f Fn(f)5 b Fr(\()p Fm(O)1784 1426 y Fk(Q)1819 1419 y Fn(=)p Fm(O)1884 1426 y Fk(P)1917 1419 y Fr(\),)0 1489 y(usually)15 b(denoted)i Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\))17 b(and)f Fn(f)5 b Fr(\()p Fn(Q=P)i Fr(\).)18 b(In)f(particular,)608 1639 y Fn(f)5 b Fr(\()p Fn(Q=P)i Fr(\))16 b(=)d([)p Fn(k)r Fr(\()p Fn(Q)p Fr(\))i(:)e Fn(k)r Fr(\()p Fn(P)7 b Fr(\)])14 b(=)1199 1605 y(deg)8 b Fn(Q)p 1199 1627 124 2 v 1199 1673 a Fr(deg)h Fn(P)1328 1639 y(:)100 1793 y Fr(By)14 b(an)g(earlier)g(result)f(\(reference!\),)802 1756 y Fh(T)844 1808 y Fk(P)5 b Fj(2)p Fk(C)943 1793 y Fm(O)983
1800 y Fk(P)1030 1793 y Fr(=)1084 1780 y(\026)1082 1793 y Fn(k)j Fm(\\)e Fn(K)19 b Fr(is)14 b(a)g(\014nite)g(algebraic)f (extension)h(of)h Fn(k)r Fr(,)0 1863 y(whic)o(h)i(w)o(e)g(can)h(assume) e(to)i(b)q(e)g(equal)f(to)h Fn(k)i Fr(\(rede\014ning)c Fn(k)j Fr(if)f(necessary\).)f(In)h(other)f(language,)f(this)0 1932 y(sa)o(ys)g(that)g(the)h(only)f(functions)f(on)h(the)h(curv)o(e)f (with)g(no)g(p)q(oles)g(are)g(the)h(constan)o(ts,)e(the)h(elemen)o(ts)g (of)0 2002 y Fn(k)r Fr(.)100 2072 y(A)h Fq(divisor)h Fn(D)g Fr(of)f Fn(C)j Fr(is)c(a)g(formal)g(\014nite)g(sum)749 2196 y Fn(D)f Fr(=)894 2149 y Fh(X)892 2249 y Fc(P)t Fi(2)p Fc(C)864 2278 y(P)20 b Fd(closed)1011 2196 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))k Fm(\001)g Fn(P)q(:)0 2389 y Fr(Sa)o(ying)21 b(the)h(sum)f(is)h(\014nite)f(means)g(that)i Fn(n)p Fr(\()p Fn(P)7 b Fr(\))23 b(=)g(0)f(for)g(all)f(but)h (\014nitely)g(man)o(y)f(p)q(oin)o(ts)g Fn(P)7 b Fr(.)22 b(The)0 2459 y(divisors)c(form)i(a)g(partially)f(order)g(ab)q(elian)g (group)g(Div)i(\(or)f(Div)1272 2466 y Fk(K)1331 2459 y Fr(if)g(the)g(\014eld)g(is)f(not)h(clear)g(from)0 2529 y(con)o(text\),)d(ordered)e(b)o(y)i(the)f(relation)392 2615 y Fh(X)473 2663 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))k Fm(\001)g Fn(P)21 b Fm(\025)721 2615 y Fh(X)802 2663 y Fn(m)p Fr(\()p Fn(P)7 b Fr(\))k Fm(\001)g Fn(P)57 b Fr(i\013)49 b Fm(8)p Fn(P)20 b Fr(:)14 b Fn(n)p Fr(\()p Fn(P)7 b Fr(\))14 b Fm(\025)f Fn(m)p Fr(\()p Fn(P)7 b Fr(\))p Fn(:)950 2790 y Fr(30)p eop %%Page: 31 31 31 30 bop 0 50 a Fr(A)15 b(divisor)e Fn(D)j Fr(is)e Fq(e\013e)m(ctive)g Fr(if)h Fn(D)g Fm(\025)f Fr(0)f Fm(\021)750 12 y Fh(P)811 50 y Fr(0)852 37 y(_)836 50 y Fn(P)7 b Fr(,)14 b(and)g(is)g Fq(prime)h Fr(if)f(there)h(is)f(a)g(p)q(oin)o(t)g Fn(P)21 b Fr(so)14 b(that)h Fn(D)h Fr(=)d Fn(P)7 b Fr(.)100 120 y(The)16 b(degree,)g(de\014ned)g(ab)q(o)o(v)o(e)g(for)g(p)q(oin)o(ts,)f (has)h(a)h(natural)e(extension)i(to)f(divisors:)590 248 y(deg\()684 200 y Fh(X)765 248 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))12 b Fm(\001)e Fn(P)d Fr(\))15 b(=)1033 200 y Fh(X)1113 248 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))h(deg)i Fn(P)q(:)100 376 y Fr(The)22 b(image)g(of)h(the)g(degree)f(map)g(is)g (a)g(subgroup)f(of)h Fp(Z)p Fr(,)h(of)g(\014nite)f(index)g(since)g(it)h (con)o(tains)0 446 y(deg)9 b Fn(P)25 b Fm(6)p Fr(=)18 b(0)h(for)g(an)o(y)g(p)q(oin)o(t)g Fn(p)p Fr(.)g(It)g(is)g(an)g(in)o (teresting)f(question)h(when)g(this)f(map)h(is)g(surjectiv)o(e.)f(F)l (or)0 515 y(instance,)e(it)i(is)e(a)h(non-trivial)f(fact)i(\(to)f(b)q (e)h(pro)o(v)o(ed)d(later)i([ref)t(]\))g(that)g(when)g Fn(k)i Fr(is)d(a)h(\014nite)g(\014eld,)f Fn(deg)0 585 y Fr(is)h(surjectiv)o(e.)g(On)g(the)h(other)f(hand,)f(it)i(is)f(easy)h (to)g(see)f(that)h(when)f Fn(k)i Fr(is)e(algebraically)g(closed,)g(deg) 0 655 y(is)f(surjectiv)o(e,)g(as)g(deg)9 b Fn(P)21 b Fr(=)13 b(1)k(for)f(an)o(y)g(p)q(oin)o(t)g Fn(p)p Fr(.)100 724 y(There)g(is)g(a)g(natural)g(group)f(homomorphism)688 843 y Fn(K)734 822 y Fj(\003)771 843 y Fm(!)e Fr(Div)913 850 y Fk(K)727 942 y Fn(f)20 b Fm(7!)13 b Fr(\()p Fn(f)5 b Fr(\))16 b Fm(\021)969 895 y Fh(X)990 1002 y Fk(P)1049 942 y Fr(ord)1121 949 y Fk(P)1155 942 y Fr(\()p Fn(f)5 b Fr(\))p Fn(P)0 1110 y Fr(\(This)19 b(is)f(a)h(\014nite)g(sum)f(b)o(y) g(earlier)g(results)g(\(ref)t(!\).\))h(The)g(image)g(of)g(this)f(map)g (is)h(called)f(the)i(set)f(of)0 1180 y Fq(princip)m(al)k(divisors)p Fr(.)f(By)f(our)e(assumption)f(ab)q(o)o(v)o(e,)h(the)i(k)o(ernel)e(is)g Fn(k)1330 1162 y Fj(\003)1353 1180 y Fr(.)h(The)g(cok)o(ernel)f(is)h (new;)f(it)i(is)0 1250 y(called)15 b(the)h Fq(Pic)m(ar)m(d)j(gr)m(oup)e
Fr(Pic)586 1257 y Fk(K)640 1250 y Fr(of)f Fn(C)j Fr(\(or)d(the)g Fq(divisor)j(class)f(gr)m(oup)f Fr(of)e Fn(C)t Fr(\);)h(it)g(is)f(an)g (in\014nite)g(group.)0 1319 y(Later)h(w)o(e)g(will)g(see)g(an)g (alternativ)o(e)g(de\014nition)f(of)h(the)h(Picard)e(group)g(as)h(the)g (group)f(of)i(equiv)m(alence)0 1389 y(classes)f(of)g(line)g(bundles,)f (with)i(sum)e(corresp)q(onding)f(to)j(the)g(tensor)f(pro)q(duct.)100 1459 y(T)l(o)g(see)g(that)h(Pic)431 1466 y Fk(K)486 1459 y Fr(is)f(in\014nite,)g(consider)f(the)i(sequence)f(of)h(maps)787 1595 y Fn(K)833 1574 y Fj(\003)870 1595 y Fm(!)d Fr(Div)1013 1602 y Fk(K)1059 1564 y Fl(deg)1064 1595 y Fm(!)g Fp(Z)0 1716 y Fr(The)21 b(comp)q(osition)f(is)g(0.)h(\(This)g(has)f(already)g (b)q(een)h(stated,)g(but)g(not)g(y)o(et)h(completely)e(pro)o(v)o(ed;)g (it)0 1786 y(dep)q(ends)d(on)h(the)h(fact)g(that)562 1749 y Fh(P)623 1786 y Fn(ef)k Fr(=)17 b Fn(n)h Fr(for)g(the)g(relev)o (en)o(t)g(\014nite)g(extensions)g(of)g(degree)g Fn(n)p Fr(.\))h(Hence)0 1856 y(the)j(degree)f(map)g(descends)g(to)h(a)g(map)f (Pic)877 1863 y Fk(K)938 1856 y Fm(!)h Fp(Z)g Fr(\(in)g(other)g (language,)e(the)i(degree)g(of)g(a)g(line)0 1926 y(bundle)c(is)h(w)o (ell-de\014ned\))f(and)h(Pic)684 1933 y Fk(K)742 1926 y Fr(is)g(in\014nite,)g(as)g(an)o(y)g(divisor)g(class)g(with)g (non-zero)f(v)m(aluation)0 1995 y(will)e(ha)o(v)o(e)g(in\014nite)g (order.)100 2065 y(F)l(or)f Fn(D)g Fm(2)g Fr(Div)371 2072 y Fk(K)410 2065 y Fr(,)h(de\014ne)g(a)h(v)o(ector)f(space)455 2183 y Fn(L)p Fr(\()p Fn(D)q Fr(\))f(=)e Fm(f)p Fr(0)p Fm(g)e([)h(f)p Fn(f)19 b Fm(2)14 b Fn(K)928 2163 y Fj(\003)965 2183 y Fr(:)f(\()p Fn(f)5 b Fr(\))16 b Fm(\025)d(\000)p Fn(D)584 2273 y Fr(=)g Fm(f)p Fr(0)p Fm(g)e([)h(f)p Fn(f)19 b Fm(2)14 b Fn(K)928 2252 y Fj(\003)965 2273 y Fr(:)f Fm(8)p Fn(P)20 b Fr(:)13 b(ord)1173 2280 y Fk(P)1214 2273 y Fn(f)20 b Fm(\025)13 b(\000)p Fn(n)p Fr(\()p Fn(P)7 b Fr(\))p Fm(g)p Fn(:)0 2391 y Fr(Wh)o(y)15 b(is)f(there)h(a)g(min)o (us)d(sign?)i(This)g(de\014nition)g(is)g(primarily)f(of)i(in)o(terest)f (when)g Fn(D)j Fr(is)d(e\013ectiv)o(e.)i(But)0 2461 y(it)g(is)g (immediate)f(that)h Fn(L)p Fr(\(0\))f(=)e Fn(k)18 b Fr(and)d(hence)h Fn(L)p Fr(\()p Fm(\000)p Fn(D)q Fr(\))f(=)f Fm(f)p Fr(0)p Fm(g)h Fr(for)h(an)o(y)g Fn(D)f(>)f Fr(0.)i(Without)g(the)g(min)o(us)0 2531 y(sign)e(in)h(the)h(de\014nition,)e(w)o(e)h(w)o(ould)f(ha)o(v)o(e) g Fn(L)p Fr(\()p Fn(D)q Fr(\))h(=)f(0)h(instead.)f(Another)h(reason)f (for)h(the)h(min)o(us)d(sign)0 2600 y(is)j(the)h(connection)f(with)g (line)g(bundles.)f(W)l(e'll)h(asso)q(ciate)h(a)f(bundle)f(to)i(eac)o(h) f Fn(D)q Fr(,)h(and)f Fn(L)p Fr(\()p Fn(D)q Fr(\))i(will)e(b)q(e)0 2670 y(iden)o(ti\014ed)f(with)i(the)f(space)h(of)f(sections)g(of)h Fn(D)q Fr(.)950 2790 y(31)p eop %%Page: 32 32 32 31 bop 100 50 a Fn(L)p Fr(\()p Fn(D)q Fr(\))15 b Fm(\032)e Fn(K)21 b Fr(is)16 b(a)g Fn(k)i Fr(v)o(ector)f(space,)f(so)g(w)o(e)h (can)f(use)g(its)g(dimension)756 219 y Fn(l)q Fr(\()p Fn(D)q Fr(\))860 177 y(def)875 206 y Fm(\030)875 221 y Fr(=)935 219 y(dim)1018 226 y Fk(K)1065 219 y Fn(L)p Fr(\()p Fn(D)q Fr(\))p Fn(:)0 349 y Fr(W)l(e)h(immediately)e(ha)o(v)o (e)h(that)h Fn(l)q Fr(\(0\))e(=)e(1.)j(It's)h(not)f(ob)o(vious)g(y)o (et,)h(but)f Fn(l)q Fr(\()p Fn(D)q Fr(\))i(turns)d(out)i(to)g(b)q(e)f (\014nite.)0 450 y Fp(Ad)o(\022)-27 b(eles.)100 520 y Fr(The)16 b(Ad)o(\023)-24 b(ele)17 b(ring)e Fp(A)487 527 y Fk(K)543 520 y Fr(of)h Fn(K)21 b Fr(is,)16 b(b)o(y)g (de\014nition,)808 665 y Fp(A)851 672 y Fk(K)904 665 y Fr(=)968 617 y Fh(Y)1032 625 y Fj(0)963 724 y Fk(P)5 b Fj(2)p Fk(C)962 753 y Fd(closed)1066 665 y Fn(K)1108 672 y Fk(P)0 878 y Fr(The)19 b(notation)307 840 y Fh(Q)354
852 y Fj(0)388 878 y Fr(means)f(the)i Fq(r)m(estricte)m(d)g Fr(direct)f(pro)q(duct.)g(It)h(needs)f(additional)f(information)g(to)0 947 y(b)q(e)f(w)o(ell-de\014ned;)d(in)i(this)g(case)h(it)g(means)420 1080 y Fp(A)463 1087 y Fk(K)515 1080 y Fr(=)d Fm(f)8 b Fr(\()p Fn(x)648 1087 y Fk(P)682 1080 y Fr(\))14 b Fm(2)762 1042 y Fh(Q)818 1080 y Fn(K)860 1087 y Fk(P)907 1080 y Fr(:)g Fn(x)963 1087 y Fk(P)1010 1080 y Fm(2)1071 1067 y Fr(^)1057 1080 y Fm(O)1097 1087 y Fk(P)1147 1080 y Fr(for)i(almost)g(all)g Fn(P)7 b Fm(g)p Fn(:)0 1210 y Fr(\(As)17 b(usual,)e(`almost)h(all')g(means)f(`all)h(but)h (\014nitely)f(man)o(y)l(.'\))100 1280 y Fp(A)143 1287 y Fk(K)198 1280 y Fr(has)g(a)g(natural)g(top)q(ology)l(.)g(W)l(e)h(ha)o (v)o(e)822 1370 y Fh(Y)838 1477 y Fk(P)907 1405 y Fr(^)894 1418 y Fm(O)934 1425 y Fk(P)980 1418 y Fm(\032)d Fp(A)1076 1425 y Fk(K)1115 1418 y Fn(:)0 1605 y Fr(Giv)o(e)120 1568 y Fh(Q)189 1592 y Fr(^)175 1605 y Fm(O)215 1612 y Fk(P)269 1605 y Fr(the)20 b(pro)q(duct)f(top)q(ology)l(.)h(Then,)g (in)f Fp(A)1018 1612 y Fk(K)1057 1605 y Fr(,)h(de\014ne)g(it)g(and)f (its)h(translates)g(to)g(b)q(e)g(op)q(en.)0 1675 y(\(Note)j(that)f (this)f(is)h(di\013eren)o(t)f(from)f(the)i(top)q(ology)g(induced)f (from)1355 1637 y Fh(Q)1410 1675 y Fn(K)1452 1682 y Fk(P)1486 1675 y Fr(.\))h(There)f(is)g(a)h(natural)0 1745 y(inclusion)g Fn(K)29 b Fm(\022)d Fp(A)394 1752 y Fk(K)456 1745 y Fr(\()p Fn(K)i Fr(is)23 b(in)667 1707 y Fh(Q)722 1745 y Fn(K)764 1752 y Fk(P)822 1745 y Fr(and)f(lands)h(in)g(the)h(restricted)f(pro)q (duct\).)g(What)h(is)f(the)0 1814 y(induced)15 b(top)q(ology)i(on)f Fn(K)t Fr(?)g(F)l(or)g(an)g(arbitrary)f Fn(P)948 1821 y Fl(0)984 1814 y Fm(2)f Fn(C)t Fr(,)i(consider)f(the)i(op)q(en)f(set) 808 1950 y(^)801 1952 y Ff(m)839 1959 y Fk(P)864 1964 y Fd(0)898 1952 y Fm(\002)969 1905 y Fh(Y)948 2012 y Fk(P)5 b Fj(6)p Fl(=)p Fk(P)1034 2017 y Fd(0)1062 1952 y Fm(O)1102 1959 y Fk(P)1135 1952 y Fn(:)0 2137 y Fr(The)17 b(in)o(tersection)g(of)h(this)f(set)g(with)h Fn(K)j Fr(is)c(just)h Fn(L)p Fr(\()p Fm(\000)p Fn(P)1059 2144 y Fl(0)1081 2137 y Fr(\))e(=)f Fm(f)8 b Fr(0)g Fm(g)p Fr(,)18 b(so)f(the)h(induced)e (top)q(ology)i(on)f Fn(K)0 2206 y Fr(is)f(the)h(discrete)f(top)q(ology) l(.)100 2276 y(An)o(y)g(divisor)f Fn(D)k Fr(giv)o(es)d(a)g(fractional)g (ideal)668 2414 y Ff(d)e Fr(=)808 2367 y Fh(Y)760 2484 y Fk(P)5 b Fr(closed)921 2414 y(\()947 2412 y(^)940 2414 y Ff(m)978 2421 y Fk(P)1011 2414 y Fr(\))1030 2393 y Fk(n)p Fl(\()p Fk(P)g Fl(\))1134 2414 y Fm(\022)13 b Fp(A)1229 2421 y Fk(K)1268 2414 y Fn(:)0 2599 y Fr(By)k(de\014nition,) 787 2670 y Fn(L)p Fr(\()p Fn(D)q Fr(\))f(=)d Fn(K)i Fm(\\)c Ff(d)1095 2649 y Fj(\000)p Fl(1)1149 2670 y Fn(:)950 2790 y Fr(32)p eop %%Page: 33 33 33 32 bop 0 50 a Fr(\(T)l(o)13 b(get)g Ff(d)193 32 y Fj(\000)p Fl(1)247 50 y Fr(,)f(put)h(a)g Fm(\000)f Fr(in)h(fron)o(t)f (of)h Fn(n)p Fr(\()p Fn(P)7 b Fr(\))13 b(in)g(the)g(de\014nition)f(of)h Ff(d)p Fr(.\))f(A)i(subset)e Ff(h)h Fr(of)g Fp(A)1609 57 y Fk(K)1660 50 y Fr(is)g(a)f Fq(fr)m(actional)0 123 y(ide)m(al)18 b Fr(if)f(there)f(is)g(a)h Fn(c)c Fm(2)h Fp(A)506 130 y Fk(K)561 123 y Fr(so)j(that)f Fn(c)p Ff(h)h Fr(is)f(an)g(op)q(en)g(ideal)g(of)1210 86 y Fh(Q)1279 110 y Fr(^)1265 123 y Fm(O)1305 130 y Fk(P)1338 123 y Fr(.)0 193 y Fq(Exer)m(cise:)i Fr(eac)o(h)e(fractional)g(ideal)g(comes) g(from)f(a)i(unique)f(divisor)f(in)h(this)g(w)o(a)o(y)l(.)100 263 y(If)21 b Fn(D)h Fr(is)f(e\013ectiv)o(e,)g(then)g Ff(d)g Fr(is)f(a)h(true)f(ideal.)h(If)g Fn(D)h Fr(is)f(prime,)e(then)i
Ff(d)g Fr(=)1554 261 y(^)1547 263 y Ff(m)1585 270 y Fk(P)1610 275 y Fd(0)1647 263 y Fm(\002)1699 226 y Fh(Q)1746 278 y Fk(P)5 b Fj(6)p Fl(=)p Fk(P)1832 283 y Fd(0)1877 250 y Fr(^)1863 263 y Fm(O)1903 270 y Fk(P)1936 263 y Fr(,)0 336 y(whic)o(h)15 b(is)h(a)h(prime)e(ideal.)h(If)g Fn(D)g Fm(\025)d Fn(D)708 318 y Fj(0)723 336 y Fr(,)j(then)h Ff(d)c Fm(\024)h Ff(d)983 318 y Fj(0)1013 336 y Fr(and)i(dim)1193 343 y Fk(K)1240 336 y Ff(d)1265 318 y Fj(0)1279 336 y Fn(=)p Ff(d)d Fr(=)h(deg)9 b Fn(D)j Fm(\000)f Fr(deg)d Fn(D)1707 318 y Fj(0)1738 336 y Fr(b)o(y)17 b(a)f(lo)q(cal)0 405 y(computation.)100 475 y(Finally)f(w)o(e)h(are)h(in)f(a)g(p)q (osition)g(to)h(state)0 545 y Fp(Riemann's)c(Theorem.)e Fr(F)l(or)g(ev)o(ery)i(divisor)e Fn(D)q Fr(,)i(the)g(n)o(um)o(b)q(ers)d Fn(l)q Fr(\()p Fn(D)q Fr(\))j(and)f Fn(c)p Fr(\()p Fn(D)q Fr(\))j(=)f(dim)1704 552 y Fk(k)1736 545 y Fp(A)1779 552 y Fk(K)1818 545 y Fn(=)p Fr(\()p Fn(K)7 b Fr(+)0 615 y Ff(d)25 597 y Fj(\000)p Fl(1)78 615 y Fr(\))17 b(are)f(\014nite)h(\(and)f Fm(\025)e Fr(0\),)i(and)g(if)h(w)o(e)f (write)h Fn(g)e Fr(=)f Fn(c)p Fr(\(0\))i(\(the)i Fq(genus)e Fr(of)h Fn(K)j Fr(o)o(v)o(er)c Fn(k)r Fr(\),)g(then)655 731 y(deg)9 b Fn(D)j Fm(\000)f Fn(l)q Fr(\()p Fn(D)q Fr(\))i(+)d Fn(c)p Fr(\()p Fn(D)q Fr(\))15 b(=)f Fn(g)e Fm(\000)f Fr(1)p Fn(:)0 848 y Fr(\(Assuming)k(still)h(that)455 835 y(\026)453 848 y Fn(k)d Fm(\\)e Fn(K)18 b Fr(=)13 b Fn(k)r Fr(;)j(if)h(not,)f(replace)g(1)h(in)f(the)h(equation)f(ab)q(o) o(v)o(e)g(b)o(y)h([)1638 835 y(\026)1637 848 y Fn(k)12 b Fm(\\)f Fn(K)18 b Fr(:)c Fn(k)r Fr(].)100 918 y(The)23 b(Ro)q(c)o(h)g(half)h(of)g(the)g(theorem)f(in)o(v)o(olv)o(es)f(\\iden)o (tifying)h(l\(D\))h(with)g(c\(D\).")g(Y)l(ou)g(can)f(see)0 987 y(a)h(formal)e(analogy)i(already:)e(in)i(the)g(de\014nition)e(of)i Fn(l)q Fr(\()p Fn(D)q Fr(\),)h(replace)e(in)o(tersection)g(with)g(sum)g (and)0 1057 y(dimension)15 b(with)h(co)q(dimension)f(to)i(get)g(the)g (de\014nition)e(of)i Fn(c)p Fr(\()p Fn(D)q Fr(\).)0 1181 y Fp(Lecture)j(15)c Fr(\(Octob)q(er)h(2,)f(1995\).)0 1251 y(Scrib)q(e:)g(Nghi)g(Nguy)o(en)100 1374 y(W)l(e)f(recall)f(the)h (notations:)g Fn(k)g Fm(\032)e Fn(K)19 b Fr(is)c(a)g(\014nitely)g (generated)f(\014eld)g(extension)h(of)g(transcendence)0 1451 y(degree)h(1.)g(W)l(e)h(assume)e Fn(K)g Fm(\\)571 1438 y Fr(\026)570 1451 y Fn(k)g Fr(=)f Fn(k)r Fr(.)h Fp(A)764 1458 y Fk(K)817 1451 y Fr(=)870 1414 y Fh(Q)917 1411 y Fi(0)917 1466 y Fk(P)22 b Fl(\(closed)15 b(p)q(oin)o(t\))h Fj(2)p Fk(C)1279 1471 y Fc(K)1322 1451 y Fn(K)1364 1458 y Fk(P)1414 1451 y Fr(\(restricted)h(pro)q(duct\).)100 1541 y Fp(A)143 1548 y Fk(K)195 1541 y Fm(\033)248 1503 y Fh(Q)295 1556 y Fk(P)346 1541 y Fr(^)-35 b Fn(m)380 1515 y Fk(n)p Fl(\()p Fk(P)5 b Fl(\))380 1555 y Fk(P)483 1541 y Fr(=)13 b Fn(d)p 535 1549 26 2 v 14 w Fm( )-8 b(!)13 b Fn(D)j Fr(=)789 1503 y Fh(P)842 1556 y Fk(P)884 1541 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))p Fn(P)57 b Fr(\()p Fn(n)p Fr(\()p Fn(P)7 b Fr(\))15 b Fm(2)f Fp(Z)30 b Fr(=)14 b(0)i(for)g(almost)g(all)g Fn(P)7 b Fr(\))100 1621 y Fn(L)p Fr(\()p Fn(D)q Fr(\))15 b(=)e Fm(f)p Fr(0)p Fm(g)e([)g(f)p Fn(f)20 b Fm(2)14 b Fn(K)573 1603 y Fj(\003)596 1621 y Fr(:)i(\()p Fn(f)5 b Fr(\))15 b Fm(\025)f(\000)p Fn(D)q Fm(g)g Fr(=)g Fn(K)h Fm(\\)c Fn(d)p 1035 1629 V -22 x Fj(\000)p Fl(1)1131 1621 y Fr(where)16 b Fn(d)p 1275 1629 V -22 x Fj(\000)p Fl(1)1368 1621 y Fr(=)1421 1584 y Fh(Q)1468 1636 y Fk(P)1519 1621 y Fr(^)-35 b Fn(m)1553 1595 y Fk(n)p Fl(\()p Fk(P)5 b Fl(\))1553 1636 y Fk(P)100
1694 y Fn(l)q Fr(\()p Fn(D)q Fr(\))15 b(=)e(dim)346 1701 y Fk(k)379 1694 y Fn(L)p Fr(\()p Fn(D)q Fr(\))68 b Fn(c)p Fr(\()p Fn(D)q Fr(\))15 b(=)e Fn(dim)817 1701 y Fk(k)841 1694 y Fp(A)884 1701 y Fk(K)923 1694 y Fn(=)p Fr(\()p Fn(K)i Fr(+)c Fn(d)p 1074 1702 V -22 x Fj(\000)p Fl(1)1154 1694 y Fr(\))100 1767 y Fp(Remark:)k Fr(If)i Fn(D)h Fr(=)e(0,)h(then)f Fn(c)p Fr(\(0\))e(=)g(dim)899 1774 y Fk(k)931 1767 y Fn(cok)r(er)q Fr(\()p Fn(K)19 b Fm(\000)-9 b(!)14 b Fp(A)1268 1774 y Fk(K)1307 1767 y Fn(=)1345 1755 y Fr(^)1332 1767 y Fm(O)g Fr(=)1439 1730 y Fh(L)1494 1782 y Fk(P)1536 1767 y Fn(K)1578 1774 y Fk(P)1611 1767 y Fn(=)1650 1755 y Fr(^)1636 1767 y Fm(O)1676 1774 y Fk(P)1709 1767 y Fr(\))100 1843 y(Also)p Fn(;)i(K)268 1850 y Fk(P)301 1843 y Fn(=)340 1831 y Fr(^)326 1843 y Fm(O)366 1850 y Fk(P)413 1830 y Fm(\030)413 1846 y Fr(=)466 1843 y Fn(K)q(=)p Fm(O)574 1850 y Fk(P)100 1953 y Fp(Riemann's)21 b(Theorem:)58 b Fr(F)l(or)18 b(all)i(divisors)e(D,)h(the)h(n)o(um)o(b)q (ers)d Fn(l)q Fr(\()p Fn(D)q Fr(\))k(and)e Fn(c)p Fr(\()p Fn(D)q Fr(\))i(are)e(\014nite,)0 2022 y(and)d(if)h(w)o(e)f(put)g Fn(c)p Fr(\()p Fn(D)q Fr(\))f(=)f Fn(g)k Fr(\(the)f Fn(g)r(enus)f Fr(of)h Fn(K)j Fr(o)o(v)o(er)c(k\))h(then)g(for)f(all)g(D)h(one)f(has) 667 2139 y Fn(deg)r(D)c Fm(\000)f Fn(l)q Fr(\()p Fn(D)q Fr(\))h(+)f Fn(c)p Fr(\()p Fn(D)q Fr(\))k(=)e Fn(g)g Fm(\000)e Fr(1)0 2256 y(.)100 2365 y Fn(P)c(r)q(oof)e Fr(:)18 b(There)e(is)g(an)g(exact)i(sequence)e(\(for)h(ev)o(ery)f(t)o (w)o(o)g(divisors)f Fn(D)h Fm(\025)d Fn(E)s Fr(\):)282 2483 y(0)g Fm(\000)-8 b(!)13 b Fn(L)p Fr(\()p Fn(D)q Fr(\))p Fn(=L)p Fr(\()p Fn(E)s Fr(\))j Fm(\000)-9 b(!)14 b Fn(d)p 775 2491 V -23 x Fj(\000)p Fl(1)854 2483 y Fn(=e)p 879 2491 24 2 v -21 x Fj(\000)p Fl(1)970 2483 y Fm(\000)-9 b(!)14 b Fr(\()p Fn(K)h Fr(+)c Fn(d)p 1190 2491 26 2 v -23 x Fj(\000)p Fl(1)1269 2483 y Fr(\))p Fn(=)p Fr(\()p Fn(K)16 b Fr(+)11 b Fn(e)p 1440 2491 24 2 v -21 x Fj(\000)p Fl(1)1516 2483 y Fr(\))j Fm(\000)-8 b(!)14 b Fr(0)100 2600 y(W)l(e)23 b(kno)o(w)g(from)g(last)g(time:)h(dim)773 2607 y Fk(k)805 2600 y Fn(d)p 805 2608 26 2 v -22 x Fj(\000)p Fl(1)885 2600 y Fn(=e)p 910 2608 24 2 v -18 x Fj(\000)p Fl(1)1012 2600 y Fr(=)h Fn(deg)r(D)17 b Fm(\000)e Fn(deg)r(E)s Fr(.)23 b(Consequen)o(tly)l(,)f(if)i Fn(D)j Fm(\025)e Fr(0)0 2670 y(then,)c(b)o(y)g(letting)h Fn(E)j Fr(=)c(0)h(on)f(the)g (left)i(arro)o(w,)c(w)o(e)j(ha)o(v)o(e)e Fn(l)q Fr(\()p Fn(D)q Fr(\))c Fm(\000)e Fn(l)q Fr(\(0\))22 b Fm(\024)g Fn(deg)r(D)16 b Fm(\000)e Fn(deg)r Fr(\(0\).)22 b(Since)950 2790 y(33)p eop %%Page: 34 34 34 33 bop 0 50 a Fn(l)q Fr(\(0\))14 b(=)g(1)f(and)f Fn(deg)r Fr(\(0\))i(=)g(0,)f(it)g(follo)o(ws)f(that)h Fn(l)q Fr(\()p Fn(D)q Fm(\024)d Fr(\))j Fr(1)t(+)t Fn(deg)r(D)i(<)e Fm(1)p Fr(.)g(F)l(or)f(an)o(y)h Fn(D)h Fr(not)f(neccessarily)p Fn(>)0 120 y Fr(0)k(one)f(can)g(c)o(ho)q(ose)g Fn(D)424 101 y Fj(0)455 120 y Fr(with)h Fn(D)611 101 y Fj(0)639 120 y Fm(\025)d Fr(0,)i Fn(D)789 101 y Fj(0)818 120 y Fm(\025)d Fn(D)q Fr(.)k(Then)f Fn(L)p Fr(\()p Fn(D)q Fr(\))f Fm(\032)f Fn(L)p Fr(\()p Fn(D)1350 101 y Fj(0)1365 120 y Fr(\))j(and)f Fn(l)q Fr(\()p Fn(D)q Fr(\))f Fn(<)e(l)q Fr(\()p Fn(D)1738 101 y Fj(0)1753 120 y Fr(\))h Fn(<)g Fm(1)p Fr(.)100 193 y(No)o(w)h(tak)o(e)i(an)e(arbitrary)g Fn(z)h Fm(2)e Fn(K)t Fm(n)p Fn(k)r Fr(.)h(W)l(e)h(write)g(\()p Fn(z)r Fr(\))f(=)e Fn(N)i Fm(\000)10 b Fn(D)q Fr(,)16
b(where)g Fn(N)21 b Fr(and)15 b Fn(D)j Fr(are)d(e\013ectiv)o(e)0 263 y(and)g(coprime.)f(Note)i(that)g Fn(k)r Fr(\()p Fn(z)r Fr(\))g(is)f(a)g(purely)g(transcenden)o(tal)f(\014eld)h(extension.)g (If)g(w)o(e)h(denote)f Fm(1)g Fr(for)0 332 y(the)h(in\014nit)o(y)f(p)q (oin)o(t)h(of)g(the)h(curv)o(e)e Fp(P)699 314 y Fl(1)738 332 y Fr(ha)o(ving)g(this)h(function)f(\014eld,)h(then)g Fn(D)f Fr(=)1533 295 y Fh(P)1585 347 y Fk(Q)p Fj(7!1)1707 332 y Fn(e)p Fr(\()p Fn(Q=)p Fm(1)p Fr(\))p Fn(:Q)p Fr(.)0 406 y(W)l(e)i(shall)e(study)i Fn(l)q Fr(\()p Fn(nD)q Fr(\))g(when)f Fn(n)e Fm(!)g(1)p Fr(.)100 480 y(Upp)q(er)i(b)q(ound)p 100 497 306 2 v 16 w(:)g(\(for)h(large)f Fn(n)p Fr(\))33 b Fn(l)q Fr(\()p Fn(nD)q Fr(\))15 b Fm(\024)f Fr(1)d(+)f Fn(n:deg)r(D)15 b Fr(=)f(1)d(+)27 b Fn(n)17 b(:)1430 442 y Fh(P)1483 495 y Fk(Q)p Fj(j1)1577 480 y Fn(e)p Fr(\()p Fn(Q=)p Fm(1)p Fr(\))p Fn(f)5 b Fr(\()p Fn(Q=)p Fm(1)p Fr(\))100 559 y(Lo)o(w)o(er)15 b(b)q(ound)p 100 567 302 2 v 16 w(:)h(Cho)q(ose)g(a)g(basis)g Fm(f)p Fn(u)822 566 y Fk(i)838 559 y Fm(g)g Fr(of)g Fn(K)21 b Fr(o)o(v)o(er)15 b Fn(k)r Fr(\()p Fn(z)r Fr(\).)i(Multiplying)e(eac)o(h)h Fn(u)1638 566 y Fk(i)1671 559 y Fr(b)o(y)g(a)g(suitable)0 629 y Fn(f)24 636 y Fk(i)55 629 y Fm(2)e Fn(k)r Fr([)p Fn(z)r Fr(])p Fm(nf)p Fr(0)p Fm(g)f Fr(w)o(e)g(can)g(ac)o(hiev)o(e)g Fn(or)q(d)697 636 y Fk(P)731 629 y Fn(u)760 636 y Fk(i)790 629 y Fm(\025)h Fr(0)f(for)g(all)h Fn(P)20 b Fr(not)14 b(dividing)e Fn(D)q Fr(.)i(Hence)g(\()p Fn(u)1611 636 y Fk(i)1628 629 y Fr(\))g Fm(\025)f(\000)p Fr(\()p Fn(a)1797 636 y Fk(i)1819 629 y Fr(+)5 b(1\))p Fn(D)0 699 y Fr(for)18 b(some)g(in)o(tegers)g Fn(a)418 706 y Fk(i)435 699 y Fr('s.)g(No)o(w)h Fn(L)p Fr(\()p Fn(nD)q Fr(\))f Fm(3)g Fn(z)855 681 y Fk(j)889 699 y Fm(\001)12 b Fn(u)944 706 y Fk(i)960 699 y Fr(,)19 b(0)e Fm(\024)h Fn(j)i Fm(\024)d Fn(n)12 b Fm(\000)g Fn(a)1308 706 y Fk(i)1338 699 y Fm(\000)g Fr(1.)19 b(All)g Fn(u)1560 706 y Fk(i)1594 699 y Fr(are)g Fn(k)r Fr(\()p Fn(z)r Fr(\)-linearly)0 768 y(indep)q(enden)o(t,)c(so)h (all)g Fn(z)451 750 y Fk(i)468 768 y Fn(u)497 775 y Fk(i)530 768 y Fr(are)g Fn(k)r Fr(-linearly)f(indep)q(enden)o(t.)g(Therefore,) 100 842 y Fn(l)q Fr(\()p Fn(nD)q Fr(\))g Fm(\025)293 804 y Fh(P)346 857 y Fk(i)362 842 y Fr(\()p Fn(n)d Fm(\000)e Fn(a)498 849 y Fk(i)515 842 y Fr(\))15 b(=)e Fm(\000)648 804 y Fh(P)701 857 y Fk(i)726 842 y Fn(a)752 849 y Fk(i)780 842 y Fr(+)d Fn(n)p Fr([)p Fn(K)t Fr(:)p Fn(k)r Fr(\()p Fn(z)r Fr(\)])100 915 y(Letting)17 b Fn(n)c Fm(7!)h(1)i Fr(lo)o(w)o(er)g(and)g(upp)q(er)f(b)q(ounds)g(imply:)100 951 y Fh(P)161 988 y Fn(e)p Fr(\()p Fn(Q=)p Fm(1)p Fr(\))p Fn(f)5 b Fr(\()p Fn(Q=)p Fm(1)p Fr(\))16 b Fm(\025)e Fr([)p Fn(K)t Fr(:)o Fn(k)r Fr(\()p Fn(z)r Fr(\)])100 1061 y(W)l(e)22 b(also)f(kno)o(w)g(from)h(previous)e(lectures)i(that)g Fm(\024)g Fr(is)f(true.)h(So)f(w)o(e)h(ha)o(v)o(e)f(equalit)o(y)l(,)h (and)f(the)0 1131 y(follo)o(wing)15 b(imp)q(ortan)o(t)h(equation:)794 1212 y Fn(deg)r(D)f Fr(=)e([)p Fn(K)t Fr(:)p Fn(k)r Fr(\()p Fn(z)r Fr(\)])0 1327 y(It)k(follo)o(ws)f(that)g Fn(deg)r Fr(\(\()p Fn(z)r Fr(\)\))g(=)d(0.)100 1403 y(Next,)21 b(for)g(t)o(w)o(o)f(divisors)f Fn(D)j Fr(and)e Fn(D)810 1384 y Fj(0)825 1403 y Fr(,)h Fn(D)h Fr(=)e(\()p Fn(f)5 b Fr(\))16 b(+)d Fn(D)1159 1384 y Fj(0)1215 1403 y Fm(\()-8 b(\))41 b Fn(d)p 1348 1411 26 2 v 21 w Fr(=)20 b Fn(f)g Fm(\001)14 b Fn(d)1552 1384 y Fj(0)p 1526 1411 40 2 v 1607 1403 a Fm(\()-8 b(\))41 b Fn(f)19 b Fm(\001)14 b Fn(d)p 1811 1411 26 2 v -23 x Fj(\000)p Fl(1)1911 1403
y Fr(=)0 1478 y Fn(d)26 1459 y Fj(0)p 0 1486 40 2 v 40 1452 a(\000)p Fl(1)111 1478 y Fr(=)-8 b Fm(\))16 b Fn(f)i Fm(\001)12 b Fn(L)p Fr(\()p Fn(D)q Fr(\))19 b(=)e Fn(L)p Fr(\()p Fn(D)560 1459 y Fj(0)574 1478 y Fr(\))h(=)-8 b Fm(\))16 b Fn(l)q Fr(\()p Fn(D)q Fr(\))j(=)e Fn(l)q Fr(\()p Fn(D)956 1459 y Fj(0)970 1478 y Fr(\).)i(Lik)o(ewise:)f Fn(D)h Fr(=)e(\()p Fn(f)5 b Fr(\))14 b(+)e Fn(D)1528 1459 y Fj(0)1560 1478 y Fr(=)-8 b Fm(\))16 b Fn(c)p Fr(\()p Fn(D)q Fr(\))i(=)f Fn(c)p Fr(\()p Fn(D)1916 1459 y Fj(0)1931 1478 y Fr(\))0 1547 y(\(w)o(e)g(sa)o(y)f Fn(l)q Fr(\()p Fn(D)q Fr(\))i(and)d Fn(c)p Fr(\()p Fn(D)q Fr(\))j(are)e(de\014ned)g (up)g(to)h(linear)e(equiv)m(alence.\))100 1621 y(F)l(or)g(the)i (divisor)e Fn(D)j Fr(ab)q(o)o(v)o(e)e(w)o(e)h(ha)o(v)o(e:)e Fn(l)q Fr(\()p Fn(nD)q Fr(\))g Fm(\025)f Fn(deg)r Fr(\()p Fn(nD)q Fr(\))e Fm(\000)1280 1583 y Fh(P)1341 1621 y Fn(a)1367 1628 y Fk(i)1384 1621 y Fr(.)100 1694 y Fn(C)t(l)q(aim)p 100 1702 143 2 v -1 w Fr(:)66 b Fn(l)q Fr(\()p Fn(E)s Fr(\))14 b Fm(\025)g Fn(deg)r(E)f Fm(\000)658 1656 y Fh(P)719 1694 y Fn(a)745 1701 y Fk(i)778 1694 y Fr(for)k(ev)o(ery)f (divisor)g Fn(E)s Fr(.)100 1767 y Fn(P)7 b(r)q(oof)e Fr(:)20 b(The)f(statemen)o(t)g(is)g(true)g(if)g Fn(E)h Fr(=)e Fn(nD)j Fr(for)e(some)f Fn(n)g Fm(\025)g Fr(0.)h(It)g(is)g(also) f(true)h(if)g Fn(E)i Fm(\024)c Fn(nD)0 1837 y Fr(\(same)j Fn(n)p Fr(\))h(b)q(ecause:)f(F)l(rom)f(the)i(exact)h(sequence)e(ab)q(o) o(v)o(e)g(w)o(e)h(had)f Fn(l)q Fr(\()p Fn(D)q Fr(\))15 b Fm(\000)e Fn(l)q Fr(\()p Fn(E)s Fr(\))21 b Fm(\024)f Fn(deg)r(D)15 b Fm(\000)f Fn(deg)r(E)0 1907 y Fr(if)21 b Fn(D)h Fm(\025)e Fn(E)s Fr(.)h(So)f Fn(l)q Fr(\()p Fn(E)s Fr(\))14 b Fm(\000)f Fn(deg)r(E)23 b Fm(\025)e Fn(l)q Fr(\()p Fn(D)q Fr(\))15 b Fm(\000)e Fn(deg)r(D)q Fr(.)21 b(F)l(or)e(general)h Fn(E)s Fr(:)g(\014nd)g Fn(f)27 b Fm(2)20 b Fn(k)r Fr([)p Fn(z)r Fr(])p Fm(nf)p Fr(0)p Fm(g)g Fr(suc)o(h)g(that)0 1976 y Fn(E)13 b Fm(\000)c Fr(\()p Fn(f)c Fr(\))16 b(=)d Fn(E)273 1958 y Fj(0)303 1976 y Fr(has)i(co)q(e\016cien)o(ts)h Fm(\024)g Fr(0)g(for)f(all)h Fn(P)23 b Fr(not)16 b(dividing)f Fn(D)q Fr(.)h(Then)g(the)g(statemen)o (t)g(is)g(true)f(for)0 2046 y Fn(E)40 2028 y Fj(0)70 2046 y Fr(\(since)i Fm(9)p Fn(n)f Fr(suc)o(h)f(that)i Fn(nD)e Fm(\025)f Fn(E)685 2028 y Fj(0)699 2046 y Fr(\))j(and)e(hence)i (for)f Fn(E)j Fr(b)q(ecause)e(E)f(is)g(linearly)g(equiv)m(alen)o(t)h (to)f Fn(E)1903 2028 y Fj(0)1917 2046 y Fr(.)0 2193 y Fp(Lecture)k(16)c Fr(\(Octob)q(er,)h(4,)f(1995\).)0 2267 y(Scrib)q(e:)g(Saul)f(Sc)o(hleimer)100 2417 y(T)l(o)q(da)o(y)h(w)o(e)g (\014nish)f(the)i(pro)q(of)f(of)h(Riemann's)e(theorem.)100 2531 y Fp(Theorem:)c Fr(Let)i Fn(k)i Fr(b)q(e)e(an)o(y)g(\014eld,)f (and)g(let)i Fn(K)j Fr(b)q(e)c(a)g(function)f(\014eld)h(o)o(v)o(er)f Fn(k)r Fr(.)g(\(i.e.)h Fn(k)i Fr(has)d(transcen-)0 2600 y(dence)h(degree)f(1)h(o)o(v)o(er)f Fn(k)i Fr(and)561 2587 y(\026)560 2600 y Fn(k)5 b Fm(\\)t Fn(K)17 b Fr(=)d Fn(k)r Fr(.\))f(Let)g Fn(D)i Fr(b)q(e)e(a)f(divisor.)g(Let)h Fn(c)p Fr(\()p Fn(D)q Fr(\))i(=)f Fn(dim)1571 2607 y Fk(k)1595 2600 y Fr(\()p Fp(A)1657 2607 y Fk(k)1682 2600 y Fn(=)p Fr(\()p Fn(K)7 b Fr(+)t Fn(d)1844 2582 y Fj(\000)p Fl(1)1897 2600 y Fr(\)\),)0 2670 y Fn(l)q Fr(\()p Fn(D)q Fr(\))15 b(=)f Fn(dim)251 2677 y Fk(k)275 2670 y Fn(L)p Fr(\()p Fn(D)q Fr(\).)k(Then)e Fn(c)p Fr(\()p Fn(D)q Fr(\))f Fm(\024)e(1)k Fr(and)e Fn(deg)r(D)e Fm(\000)e Fn(l)q Fr(\()p Fn(D)q Fr(\))h(+)f Fn(c)p Fr(\()p Fn(D)q Fr(\))j(=)g Fn(g)f Fm(\000)d Fr(1)17 b(where)f Fn(g)f
Fr(=)f Fn(c)p Fr(\(0\).)950 2790 y(34)p eop %%Page: 35 35 35 34 bop 100 50 a Fp(Pro)r(of:)14 b Fr(\(con)o(t\))j(Let)h Fn(D)d Fm(\025)f Fn(E)s Fr(.)i(Last)g(time)h(w)o(e)f(used)g(the)h (short)e(exact)j(sequence)342 172 y(0)c Fm(!)g Fn(L)p Fr(\()p Fn(D)q Fr(\))p Fn(=L)p Fr(\()p Fn(E)s Fr(\))h Fm(!)f Fn(d)801 152 y Fj(\000)p Fl(1)854 172 y Fn(=e)902 152 y Fj(\000)p Fl(1)970 172 y Fm(!)f Fr(\()p Fn(K)j Fr(+)10 b Fn(d)1185 152 y Fj(\000)p Fl(1)1239 172 y Fr(\))p Fn(=)p Fr(\()p Fn(K)15 b Fr(+)c Fn(e)1432 152 y Fj(\000)p Fl(1)1486 172 y Fr(\))j Fm(!)g Fr(0)0 295 y(to)i(sho)o(w)f(that)h Fn(l)q Fr(\()p Fn(D)q Fr(\))11 b Fm(\000)f Fn(l)q Fr(\()p Fn(E)s Fr(\))k Fm(\024)f Fn(deg)r Fr(\()p Fn(D)q Fr(\))e Fm(\000)f Fn(deg)r Fr(\()p Fn(E)s Fr(\);)15 b(and,)g(if)h Fn(z)h Fm(2)d Fn(K)f Fm(\000)d Fn(k)17 b Fr(then)f(w)o(e)g(ha)o(v)o(e)f (the)h(equalit)o(y)619 379 y Fh(X)699 426 y Fn(e)p Fr(\()p Fn(Q=)p Fm(1)p Fr(\))p Fn(f)5 b Fr(\()p Fn(Q=)p Fm(1)p Fr(\))17 b(=)c([)p Fn(K)18 b Fr(:)c Fn(k)r Fr(\()p Fn(z)r Fr(\)])0 555 y(where)22 b(the)h(sum)e(runs)g(o)o(v)o(er)h(all)g Fn(Q)i Fm(2)g Fn(C)814 562 y Fk(K)874 555 y Fr(suc)o(h)d(that)i Fn(Q)h Fm(7!)g(1)e Fr(under)f Fn(z)r Fr(.)i(Also,)f Fm(8)p Fn(z)j Fm(2)f Fn(K)1836 537 y Fj(\003)1881 555 y Fr(the)0 625 y Fn(deg)r Fr(\(\()p Fn(z)r Fr(\)\))15 b(=)f(0.)i(And)h(\014nally)l (,)e(w)o(e)h(sho)o(w)o(ed)f(that)i Fm(8)p Fn(E)s Fr(,)e Fn(l)q Fr(\()p Fn(E)s Fr(\))f Fm(\025)g Fn(deg)r Fr(\()p Fn(E)s Fr(\))d Fm(\000)1395 587 y Fh(P)1455 625 y Fn(a)1481 632 y Fl(1)1521 625 y Fr(\()p Fn(a)1566 632 y Fk(i)1597 625 y Fm(\025)i(\000)p Fr(1\).)100 694 y(It)18 b(is)g(no)o(w)f(easy)i (to)f(sho)o(w)f(that)i Fn(c)p Fr(\()p Fn(D)q Fr(\))e Fm(\024)f(1)i Fr(b)o(y)g(v)m(arying)g Fn(D)q Fr(.)h Fm(8)p Fn(D)e Fm(\025)f Fn(E)21 b Fr(w)o(e)d(ha)o(v)o(e)f(the)i(string)e(of)0 764 y(inequalities:)416 834 y Fn(dim)503 841 y Fk(k)527 834 y Fr(\()p Fn(K)f Fr(+)11 b Fn(d)680 813 y Fj(\000)p Fl(1)733 834 y Fr(\))p Fn(=)p Fr(\()p Fn(K)16 b Fr(+)10 b Fn(e)926 813 y Fj(\000)p Fl(1)980 834 y Fr(\))k(=)g Fn(dim)p Fr(\()p Fn(d)1198 813 y Fj(\000)p Fl(1)1251 834 y Fr(\))e Fm(\000)f Fn(dim)p Fr(\()p Fn(e)1461 813 y Fj(\000)p Fl(1)1515 834 y Fr(\))607 935 y Fm(\024)j Fn(deg)r Fr(\()p Fn(D)q Fr(\))e Fm(\000)f Fn(deg)r Fr(\()p Fn(E)s Fr(\))g Fm(\000)g Fn(l)q Fr(\()p Fn(D)q Fr(\))h(+)f Fn(l)q Fr(\()p Fn(E)s Fr(\))703 1046 y Fm(\024)755 999 y Fh(X)835 1046 y Fn(a)861 1053 y Fk(i)889 1046 y Fm(\000)g Fn(deg)r Fr(\()p Fn(E)s Fr(\))g(+)g Fn(l)q Fr(\()p Fn(E)s Fr(\))0 1160 y(No)o(w,)128 1123 y Fh(P)188 1160 y Fn(a)214 1167 y Fk(i)243 1160 y Fm(\000)g Fn(deg)r Fr(\()p Fn(E)s Fr(\))g(+)g Fn(l)q Fr(\()p Fn(E)s Fr(\))17 b(is)f(\014nite,)h(and)f(so) g(ev)o(ery)h Fn(k)r Fr(-v)o(ector)f(space)g(\()p Fn(K)g Fr(+)11 b Fn(d)1580 1142 y Fj(\000)p Fl(1)1633 1160 y Fr(\))p Fn(=)p Fr(\()p Fn(K)16 b Fr(+)11 b Fn(e)1827 1142 y Fj(\000)p Fl(1)1880 1160 y Fr(\))18 b(is)0 1230 y(\014nite)e(dimensional.)e(Th)o(us)h(\()560 1192 y Fh(S)601 1230 y Fr(\()p Fn(K)g Fr(+)10 b Fn(d)752 1212 y Fj(\000)p Fl(1)806 1230 y Fr(\)\))p Fn(=)p Fr(\()p Fn(K)15 b Fr(+)10 b Fn(e)1017 1212 y Fj(\000)p Fl(1)1071 1230 y Fr(\))16 b(is)g(as)g(w)o(ell.)f(Note)j(ho)o(w)o(ev)o(er,)1656 1192 y Fh(S)1698 1230 y Fr(\()p Fn(d)1743 1212 y Fj(\000)p Fl(1)1796 1230 y Fr(\))d(=)e Fp(A)1925 1237 y Fk(k)0 1299 y Fr(and)j(th)o(us)f(\()225 1262 y Fh(S)267 1299 y Fr(\()p Fn(K)h Fr(+)10 b Fn(d)419 1281 y Fj(\000)p
Fl(1)473 1299 y Fr(\)\))p Fn(=)p Fr(\()p Fn(K)16 b Fr(+)11 b Fn(e)686 1281 y Fj(\000)p Fl(1)739 1299 y Fr(\))k(=)e Fp(A)868 1306 y Fk(k)893 1299 y Fn(=)p Fr(\()p Fn(K)i Fr(+)c Fn(e)1067 1281 y Fj(\000)p Fl(1)1121 1299 y Fr(\).)100 1369 y(=)-8 b Fm(\))15 b Fn(c)p Fr(\()p Fn(E)s Fr(\))i(is)f(\014nite.) 100 1439 y(Then)g(w)o(e)g(ha)o(v)o(e)g(the)h(equalit)o(y)l(,)461 1561 y Fn(deg)r Fr(\()p Fn(D)q Fr(\))c Fm(\000)d Fn(deg)r Fr(\()p Fn(E)s Fr(\))k(=)g Fn(l)q Fr(\()p Fn(D)q Fr(\))e Fm(\000)f Fn(l)q Fr(\()p Fn(E)s Fr(\))g(+)g Fn(c)p Fr(\()p Fn(E)s Fr(\))g Fm(\000)g Fn(c)p Fr(\()p Fn(D)q Fr(\))p Fn(:)0 1684 y Fr(So)17 b(for)f(an)o(y)h Fn(D)f Fm(\025)f Fn(E)s Fr(,)h Fn(deg)d Fm(\000)f Fn(l)g Fr(+)f Fn(c)17 b Fr(is)f(the)i(same)e(for)h Fn(D)h Fr(and)f Fn(E)s Fr(,)f(i.e.)h (constan)o(t.)f(Th)o(us)g Fn(deg)d Fm(\000)e Fn(l)i Fr(+)e Fn(c)17 b Fr(is)0 1753 y(constan)o(t)f(for)g(all)g Fn(D)q Fr(.)h(Letting)g Fn(D)f Fr(=)d(0)k(w)o(e)f(\014nd)g(that)g Fn(deg)r Fr(\(0\))c Fm(\000)e Fn(l)q Fr(\(0\))i(+)f Fn(c)p Fr(\(0\))j(=)f(0)e Fm(\000)g Fr(1)g(+)f Fn(c)p Fr(\(0\))k(=)g Fn(g)e Fm(\000)f Fr(1)0 1823 y(and)16 b(this)g(completes)g(the)h(pro)q (of.)100 1893 y(\(Notice)d(that)f(w)o(e)g(did)f(not)h(really)g(use)f (the)h(assumption)1172 1880 y(\026)1171 1893 y Fn(k)5 b Fm(\\)t Fn(K)18 b Fr(=)c Fn(k)r Fr(.)e(Indeed,)g(if)h(w)o(e)g(assume) f(that)0 1963 y([)15 1950 y(\026)14 1963 y Fn(k)g Fm(\\)g Fn(K)17 b Fr(:)d Fn(k)r Fr(])f Fn(<)h Fm(1)i Fr(then)h(the)f(theorem)g (b)q(ecomes)g(simply)g Fn(deg)r(D)c Fm(\000)f Fn(l)q Fr(\()p Fn(D)q Fr(\))h(+)f Fn(c)p Fr(\()p Fn(D)q Fr(\))k(=)f Fn(g)e Fm(\000)f Fr([)1716 1950 y(\026)1715 1963 y Fn(k)h Fm(\\)g Fn(K)17 b Fr(:)d Fn(k)r Fr(]\))100 2072 y(Another)d(statemen)o (t)g(of)h(the)g(theorem)f(is:)g(F)l(or)f(ev)o(ery)i Fn(D)j Fm(\025)f Fn(E)g Fr(one)e(has)f Fm(9)p Fn(D)i Fr(suc)o(h)d(that)i Fn(K)5 b Fr(+)q Fn(d)1844 2054 y Fj(\000)p Fl(1)1911 2072 y Fr(=)0 2142 y Fp(A)43 2149 y Fk(K)98 2142 y Fr(and)16 b(this)g(is)g(equiv)m(alen)o(t)h(to)g(sa)o(ying)e Fn(c)p Fr(\()p Fn(D)q Fr(\))g(=)f(0.)100 2252 y Fp(Example:)g Fr(W)l(e)h(will)f(\014nd)g Fn(g)j Fr(for)d(the)i(function)e(\014eld)g Fn(K)k Fr(=)c Fn(k)r Fr(\()p Fn(z)r Fr(\).)h(Let)h Fn(u)1471 2259 y Fk(i)1500 2252 y Fr(=)e(1.)h(\(Recall)g(that)g(the)0 2321 y Fn(u)29 2328 y Fk(i)45 2321 y Fr('s)g(w)o(ere)g(to)h(b)q(e)g(a)g (basis)e(for)i Fn(K)j Fr(o)o(v)o(er)c Fn(k)r Fr(\()p Fn(z)r Fr(\),)h(where)g Fn(z)i Fr(w)o(as)d(an)o(y)g(su\016cien)o(tly)f (in)o(teresting)h(elemen)o(t)g(of)0 2391 y Fn(K)10 b Fm(\000)c Fn(k)r Fr(.\))13 b(Note)h(that)g Fn(u)422 2398 y Fk(i)452 2391 y Fr(has)g(no)f(p)q(oles.)g(W)l(e)h(ha)o(v)o(e)g Fn(a)964 2398 y Fk(i)994 2391 y Fm(2)g Fp(Z)g Fr(suc)o(h)f(that)h(the)g (divisor)f(\()p Fn(u)1596 2398 y Fk(i)1612 2391 y Fr(\))i Fm(\025)e(\000)p Fr(\()p Fn(a)1782 2398 y Fk(i)1805 2391 y Fr(+)6 b(1\))p Fn(D)q Fr(,)0 2461 y(where)15 b Fn(D)i Fr(is)f(the)g(denominator)d(of)j Fn(z)r Fr(.)g(The)g(smallest)e(p)q (ossible)h(c)o(hoice)g(for)g Fn(a)1463 2468 y Fk(i)1496 2461 y Fr(is)g Fm(\000)p Fr(1,)g(and)g(so)h(w)o(e)f(\014nd)0 2531 y(that)i(0)d Fm(\024)f Fn(g)j Fm(\024)d Fr(0,)k(i.e.)f Fn(g)f Fr(=)f(0.)100 2600 y(Note)k(that)f(it)h(is)e(also)h(p)q(ossible) f(to)h(compute)g Fn(g)i Fr(using)d Fn(c)p Fr(\(0\))f(=)f Fn(g)r Fr(.)j(Let)h Fn(L)f Fr(b)q(e)g(a)g(\014nite)g(extension)0 2670 y(of)g Fn(K)t Fr(.)f(Then)g(for)g(an)o(y)g(closed)g(p)q(oin)o(t)g Fn(P)21 b Fm(2)14 b Fn(C)846 2677 y Fk(K)901 2670 y Fr(there)i(are)g(p) q(oin)o(ts)g Fn(Q)1300 2677 y Fk(i)1331 2670 y Fm(2)e Fn(C)1414 2677 y Fk(L)1460 2670 y Fr(lying)i(o)o(v)o(er)g Fn(p)p Fr(.)950 2790 y(35)p eop
%%Page: 36 36 36 35 bop 100 50 a Fp(Claim:)281 12 y Fh(P)334 65 y Fk(Q)365 70 y Fc(i)391 50 y Fn(e)p Fr(\()p Fn(Q)472 57 y Fk(i)490 50 y Fn(=P)7 b Fr(\))p Fn(f)e Fr(\()p Fn(Q)660 57 y Fk(i)678 50 y Fn(=P)i Fr(\))14 b(=)g([)p Fn(L)f Fr(:)h Fn(K)t Fr(])100 122 y Fp(Pro)r(of:)j Fr(Pic)o(k)i Fn(z)h Fm(2)e Fn(K)t Fr(,)h Fn(or)q(d)634 129 y Fk(P)668 122 y Fr(\()p Fn(z)r Fr(\))g Fn(<)f Fr(0)h(i.e.)g Fn(z)j Fr(has)c(a)h(p)q(ole)g(at)g Fn(P)7 b Fr(.)19 b(By)h(our)e(pro)q(of)h(of)g(Riemann's)0 192 y(Theorem)c(w)o(e)i(kno)o(w)f(that)625 255 y Fh(X)544 362 y Fk(Q)p Fj(7!12)p Fk(C)711 369 y Fc(k)p Fd(\()p Fc(z)q Fd(\))786 303 y Fn(e)p Fr(\()p Fn(Q=)p Fm(1)p Fr(\))p Fn(f)5 b Fr(\()p Fn(Q=)p Fm(1)p Fr(\))16 b(=)e([)p Fn(L)f Fr(:)h Fn(k)r Fr(\()p Fn(z)r Fr(\)])0 476 y(for)j Fn(Q)e Fm(2)g Fn(C)215 483 y Fk(L)244 476 y Fr(.)i(\(Actually)l(,)g (this)g(equalit)o(y)g(do)q(es)g(not)g(directly)g(follo)o(w)f(when)h Fn(L)11 b Fm(\\)1544 463 y Fr(\026)1543 476 y Fn(k)16 b Fm(6)p Fr(=)e Fn(k)r Fr(.)i(This)h(case)g(is)0 546 y(left)f(as)f(an)g(exercise.\))h(No)o(w,)f(since)678 508 y Fh(P)744 546 y Fr(=)f(\()816 508 y Fh(P)869 561 y Fk(P)899 551 y Fi(0)913 561 y Fj(7!1)1003 546 y Fn(ef)5 b Fr(\()p Fn(P)1113 528 y Fj(0)1128 546 y Fn(=)p Fm(1)p Fr(\)\)\()1260 508 y Fh(P)1314 561 y Fk(Q)p Fj(7!)p Fk(P)1415 551 y Fi(0)1440 546 y Fn(ef)g Fr(\()p Fn(Q=P)1614 528 y Fj(0)1629 546 y Fr(\)\))16 b(and)f(the)h(\014rst)0 618 y(sum)f(on)i(the)f(righ)o(t)g(hand)f(side)h(is)g(equal)h(to)g([)p Fn(K)g Fr(:)d Fn(k)r Fr(\()p Fn(z)r Fr(\)],)j(the)g(desired)e(equalit)o (y)h(holds.)0 727 y Fp(Remark:)e Fr(F)l(rom)f(algebraic)h(geometry)g(w) o(e)h(ha)o(v)o(e)f(another)h(w)o(a)o(y)f(to)h(de\014ne)g Fn(g)r Fr(,)f(the)h(gen)o(us)f(of)h(a)g(curv)o(e.)0 797 y(W)l(e)g(set)g Fn(g)h Fr(=)d Fn(dim)339 804 y Fk(k)364 797 y Fr(\()p Fn(H)428 779 y Fl(1)451 797 y Fr(\()p Fn(C)q(;)8 b(O)567 804 y Fk(C)600 797 y Fr(\)\).)16 b(T)l(o)f(sho)o(w)e(that)j (this)e(is)h(equiv)m(alen)o(t)g(to)g(our)f(previous)g(de\014nition)g (of)0 867 y Fn(g)k Fr(consider)d(the)i(short)f(exact)h(sequence,)680 978 y(0)d Fm(!)g(O)823 985 y Fk(C)870 978 y Fm(!)f(K)i(!)e(K)q Fn(=)p Fm(O)1153 985 y Fj(C)1192 978 y Fm(!)h(0)0 1089 y Fr(where)h Fm(K)i Fr(is)e(the)h(constan)o(t)f(sheaf)g Fn(K)20 b Fr(on)15 b Fn(C)t Fr(.)g(T)l(aking)g(global)g(sections,)g(w)o (e)g(ha)o(v)o(e)g(that)h Fn(H)1704 1071 y Fl(1)1727 1089 y Fr(\()p Fn(C)q(;)8 b Fm(O)1845 1096 y Fk(C)1878 1089 y Fr(\))14 b(=)0 1159 y Fn(cok)r(er)q Fr(\()p Fn(K)23 b Fm(7!)18 b Fn(H)317 1141 y Fl(0)340 1159 y Fr(\()p Fm(K)q Fn(=)p Fm(O)463 1166 y Fj(C)489 1159 y Fr(\)\).)i(But)g Fn(H)708 1141 y Fl(0)730 1159 y Fm(K)q Fn(=)p Fm(O)834 1166 y Fj(C)878 1159 y Fr(=)935 1122 y Fh(P)996 1159 y Fm(K)q Fn(=)p Fm(O)1100 1166 y Fj(P)1152 1159 y Fr(=)e Fp(A)1252 1166 y Fj(K)1285 1159 y Fn(=)1320 1147 y Fm(^)1310 1159 y(O)q Fr(.)i(Th)o(us)e Fn(g)i Fr(=)e Fn(dim)1702 1166 y Fk(k)1726 1159 y Fr(\()p Fp(A)1788 1166 y Fk(K)1828 1159 y Fn(=K)e Fr(+)14 1219 y(^)0 1232 y Fm(O)q Fr(\))e(=)g Fn(c)p Fr(\(0\).)100 1341 y(Scrib)q(e's)g(note)i(-)f(A)o(t)h(this)f(p)q (oin)o(t)g(a)h(homew)o(ork)e(problem)g(\(5B\))i(w)o(as)f(discussed)f (at)i(some)f(length.)0 1411 y(As)h(I)h(w)o(as)f(not)g(enrolled)f(-)h (i.e.)g(not)h(doing)e(the)i(homew)o(ork)e(-)h(I)h(can't)f(do)g(justice) h(to)f(the)h(this)f(part)g(of)0 1481 y(the)j(lecture.)g(Please)g(con)o (tact)g(Professor)f(Lenstra)g(for)h(a)g(cop)o(y)g(of)g(the)h(relev)m (an)o(t)f(homew)o(ork)e(set)j(or)0 1550 y(Ho)o(w)o(ard)15 b(Thompson)g(for)h(solutions.)0 1672 y Fp(Lecture)k(17)c
Fr(\(Octob)q(er,)h(6,)f(1995\).)0 1742 y(Scrib)q(e:)g(Saul)f(Sc)o (hleimer)100 1863 y(T)l(o)q(da)o(y)h(w)o(e)g(will)g(b)q(egin)g(to)g (deriv)o(e)g(the)g(Riemann-Ro)q(c)o(h)e(Theorem)h(from)h(Riemann`s)e (Theorem)0 1933 y(pro)o(v)o(ed)i(in)g(lecture)h(16.)g(As)g(usual)f(let) i Fn(k)h Fr(b)q(e)e(a)g(\014eld)g(\(assumed)e(algebraically)h(closed)h (as)g(necessary\))0 2002 y(with)f Fn(K)21 b Fr(a)16 b(function)h (\014eld)e(o)o(v)o(er)h Fn(k)r Fr(.)100 2072 y(Recall)i(the)h(statemen) o(t)g(of)f(Riemann's)f(Theorem)h(-)g(If)h Fn(D)i Fr(is)d(a)h(divisor,)e (then)i(letting)g Fn(c)p Fr(\()p Fn(D)q Fr(\))f(=)0 2142 y Fn(dim)87 2149 y Fk(k)111 2142 y Fr(\()p Fp(A)173 2149 y Fk(k)199 2142 y Fn(=)p Fr(\()p Fn(K)d Fr(+)c Fn(d)376 2124 y Fj(\000)p Fl(1)429 2142 y Fr(\)\),)17 b Fn(L)p Fr(\()p Fn(D)q Fr(\))f(=)d Fn(K)i Fm(\\)d Fr(\()p Fn(d)827 2124 y Fj(\000)p Fl(1)880 2142 y Fr(\),)17 b(and)f Fn(l)q Fr(\()p Fn(D)q Fr(\))f(=)f Fn(dim)1278 2149 y Fk(k)1302 2142 y Fn(L)p Fr(\()p Fn(D)q Fr(\))k(w)o(e)f(ha)o(v)o(e)f(that)h Fn(c)p Fr(\()p Fn(D)q Fr(\))d Fm(\024)g(1)0 2212 y Fr(and)i Fn(deg)r(D)c Fm(\000)f Fn(l)q Fr(\()p Fn(D)q Fr(\))h(+)f Fn(c)p Fr(\()p Fn(D)q Fr(\))k(=)e Fn(g)g Fm(\000)e Fr(1)16 b(where)g Fn(g)g Fr(=)d Fn(c)p Fr(\(0\).)100 2321 y(W)l(e)j(will)h (need)f(the)h(follo)o(wing)e(useful)h(facts:)100 2391 y Fm(\017)i Fr(If)g Fn(deg)r(D)f(<)f Fr(0)i(then)g Fn(l)q Fr(\()p Fn(D)q Fr(\))g(=)d(0.)j(\(T)l(o)g(see)g(this:)g(supp)q(ose)f (that)h Fn(f)k Fm(2)16 b Fn(L)p Fr(\()p Fn(D)q Fr(\))k(and)d Fn(f)22 b Fm(6)p Fr(=)16 b(0.)i(Then)0 2461 y(\()p Fn(f)5 b Fr(\))15 b Fm(\025)f(\000)p Fn(D)q Fr(.)j(T)l(aking)f(degrees)g(w)o (e)g(ha)o(v)o(e)g(that)h(0)c Fm(\025)h(\000)p Fn(deg)r Fr(\()p Fn(D)q Fr(\),)j(i.e.)f Fn(deg)r Fr(\()p Fn(D)q Fr(\))f Fm(\025)f Fr(0.\))100 2531 y Fm(\017)h Fr(If)h Fn(deg)r Fr(\()p Fn(D)q Fr(\))f Fm(\025)f Fr(0)h(then)h Fn(l)q Fr(\()p Fn(D)q Fr(\))f Fm(\024)f Fn(deg)r Fr(\()p Fn(D)q Fr(\))c(+)f(1.)16 b(\(W)l(e)g(ha)o(v)o(e)f(already)g(dealt)h (with)f(the)h(case)g Fn(D)g Fm(\025)d Fr(0)0 2600 y(\(see)18 b(lecture)g(15.\))f(In)h(general,)f(if)g Fn(l)q Fr(\()p Fn(D)q Fr(\))g Fm(\025)f Fr(0)i(then)f Fm(9)p Fn(f)k Fm(2)16 b Fn(L)p Fr(\()p Fn(D)q Fr(\))j(suc)o(h)e(that)h Fn(f)k Fm(6)p Fr(=)15 b(0.)j(No)o(w)f(replace)g Fn(D)0 2670 y Fr(b)o(y)f(the)h(equiv)m(alen)o(t)g(divisor)e Fn(D)e Fr(+)d(\()p Fn(f)5 b Fr(\))16 b Fm(\025)d Fr(0)k(and)f(w)o(e)g (are)g(done.\))950 2790 y(36)p eop %%Page: 37 37 37 36 bop 100 50 a Fm(\017)16 b Fn(c)p Fr(\()p Fn(D)q Fr(\))f Fm(\025)e Fr(0)100 159 y Fp(De\014nition:)21 b Fr(A)e Fq(Weil)i(di\013er)m(ential)f Fr(is)f(a)g(con)o(tin)o(uous,)e Fn(k)r Fr(-linear)h(function)h Fn(\025)g Fr(:)g Fp(A)1686 166 y Fk(K)1743 159 y Fm(!)g Fn(k)i Fr(that)0 229 y(v)m(anishes)d(on)h Fn(K)k Fr(\()p Fn(k)e Fr(is)e(giv)o(en)g(the)g(discrete)g(top)q(ology)g (here.\))g(Note)h(that)f(the)h(k)o(ernal)e(of)h Fn(\025)h Fr(is)e(op)q(en)0 308 y(and)e(th)o(us)f(con)o(tains)h(some)g Fn(d)g Fr(\(=)641 271 y Fh(Q)688 323 y Fk(P)739 308 y Fr(^)-34 b Fn(m)774 282 y Fk(n)p Fl(\()p Fk(P)5 b Fl(\))774 323 y Fk(P)862 308 y Fr(\))17 b(since)f(these)h Fn(d)p Fr('s)f(are)g(basic)g(op)q(en)g(sets.)100 378 y Fp(Notation:)g Fn(D)q(if)5 b(f)21 b Fr(=)14 b Fn(D)q(if)5 b(f)652 385 y Fk(K)706 378 y Fr(=)13 b Fm(f)k Fr(all)e(W)l(eil)h(di\013eren)o (tials)f Fm(g)p Fr(.)h(This)f(is)h(a)g Fn(K)t Fr(-v)o(ector)f(space)h (with)0 448 y(scaler)k(m)o(ultiplication)e(giv)o(en)i(b)o(y)g Fn(c\025)p Fr(\()p Fn(x)p Fr(\))h(=)f Fn(\025)p Fr(\()p Fn(cx)p Fr(\))h(and)f(v)o(ector)g(addition)f(giv)o(en)h(b)o(y)g(the)h
(standard)0 517 y(p)q(oin)o(t-wise)15 b(addition)h(of)g(functions.)100 627 y(The)g(rest)g(of)h(this)f(lecture)h(will)f(b)q(e)g(dev)o(oted)h (to)g(sho)o(wing)e(that)i Fn(dim)1410 634 y Fk(K)1448 627 y Fn(D)q(if)5 b(f)21 b Fr(=)13 b(1:)100 737 y(W)l(e)22 b(note)f(that)h Fn(c)p Fr(\()p Fn(D)q Fr(\))i(=)e Fn(dim)693 744 y Fk(k)717 737 y Fr(\()p Fp(A)779 744 y Fk(K)818 737 y Fn(=)p Fr(\()p Fn(K)d Fr(+)14 b Fn(d)1002 718 y Fj(\000)p Fl(1)1055 737 y Fr(\)\))23 b(=)f Fn(dim)1264 744 y Fk(k)1289 737 y Fm(f)p Fn(f)28 b Fr(:)22 b Fp(A)1445 744 y Fk(K)1506 737 y Fm(!)g Fn(k)i Fm(j)e Fn(f)27 b Fr(is)21 b Fn(k)r Fr(-linear,)0 806 y Fn(f)32 b Fm(j)70 813 y Fk(K)108 806 y Fr(=)26 b(0,)e(and)g Fn(d)367 788 y Fj(\000)p Fl(1)446 806 y Fm(\032)i Fn(k)r(er)q Fr(\()p Fn(f)5 b Fr(\).)26 b Fm(g)e Fr(=)g Fn(dim)891 813 y Fk(k)915 806 y Fn(D)q(if)5 b(f)1027 813 y Fk(D)1090 806 y Fr(where)23 b Fn(D)q(if)5 b(f)1353 813 y Fk(D)1415 806 y Fr(is)24 b(de\014ned)f(to)h(b)q(e)h(the)f(set)0 876 y Fm(f)p Fn(\025)14 b Fm(2)g Fn(D)q(if)5 b(f)21 b Fm(j)14 b Fn(d)302 858 y Fj(\000)p Fl(1)369 876 y Fm(\032)f Fn(k)r(er)q Fr(\()p Fn(\025)p Fr(\))p Fm(g)p Fr(.)18 b(Of)f(course)e Fn(D)q(if)5 b(f)954 883 y Fk(D)1009 876 y Fr(is)16 b(a)h(sub)e Fn(k)r Fr(-v)o(ector)h(space)g(of)h Fn(D)q(if)5 b(f)g Fr(.)100 946 y(No)o(w,)13 b(if)h Fn(deg)r Fr(\()p Fn(D)q Fr(\))h Fn(<)e(g)7 b Fm(\000)e Fr(1)13 b(then)h(Riemann's)d(Theorem)i(implies)f (that)i Fn(c)p Fr(\()p Fn(D)q Fr(\))h Fn(>)e Fr(0.)h(Th)o(us)e Fn(D)q(if)5 b(f)21 b Fm(6)p Fr(=)0 1016 y(0)c(\(i.e.)f Fn(dim)228 1023 y Fk(K)266 1016 y Fn(D)q(if)5 b(f)21 b Fm(\025)14 b Fr(1.\))j(Let)g Fn(D)q Fr(,)p Fn(E)j Fr(b)q(e)d (divisors.)d(W)l(e)100 1125 y Fp(Claim:)i Fn(L)p Fr(\()p Fn(D)10 b Fm(\000)e Fn(E)s Fr(\))g Fm(\001)g Fn(D)q(if)d(f)632 1132 y Fk(D)684 1125 y Fm(\032)14 b Fn(D)q(if)5 b(f)849 1132 y Fk(E)884 1125 y Fr(.)15 b(T)l(o)g(pro)o(v)o(e)f(this)h(let)h Fn(f)j Fm(2)14 b Fn(L)p Fr(\()p Fn(D)c Fm(\000)e Fn(E)s Fr(\))16 b(and)e Fn(\025)g Fm(2)g Fn(D)q(if)5 b(f)1898 1132 y Fk(D)1936 1125 y Fr(.)0 1196 y(Then)14 b Fn(\025)g Fm(j)185 1175 y Fj(\000)p Fl(1)185 1211 y Fk(d)254 1196 y Fr(is)g(zero)h(and)f Fn(f)19 b Fm(2)c Fn(e)8 b Fm(\001)g Fn(d)671 1178 y Fj(\000)p Fl(1)724 1196 y Fr(.)14 b(Fix)h Fn(x)f Fm(2)g Fn(e)952 1178 y Fj(\000)p Fl(1)1006 1196 y Fr(.)h(Then)f Fn(f)5 b(\025)p Fr(\()p Fn(x)p Fr(\))16 b(=)e Fn(\025)p Fr(\()p Fn(f)5 b(x)p Fr(\))15 b(=)f(0)h(since)f Fn(f)g Fm(\001)8 b Fn(x)14 b Fm(2)g Fn(d)1883 1178 y Fj(\000)p Fl(1)1936 1196 y Fr(.)0 1266 y(Since)i(w)o(e)g(ma)o(y)g(c)o (hose)g Fn(f)k Fm(6)p Fr(=)13 b(0)k(w)o(e)f(are)g(done.)100 1375 y Fp(Claim:)h Fr(If)g Fn(deg)r Fr(\()p Fn(D)q Fr(\))e Fn(>)e Fr(2)p Fn(g)g Fm(\000)e Fr(2)16 b(then)h Fn(c)p Fr(\()p Fn(D)q Fr(\))d(=)g(0.)100 1445 y(Let)e Fn(E)j Fr(b)q(e)d(an)o(y)f(divisor)g(with)h Fn(l)q Fr(\()p Fn(E)s Fr(\))i(=)f(0.)f(Supp)q(ose)e(that)j Fn(c)p Fr(\()p Fn(D)q Fr(\))h Fn(>)g Fr(0)e(Then)f Fn(c)p Fr(\()p Fn(D)q Fr(\))k(=)f Fn(dimD)q(if)5 b(f)1860 1452 y Fk(D)1911 1445 y Fn(>)0 1515 y Fr(0)17 b(implies)e(that)293 1641 y Fn(dim)380 1648 y Fk(k)404 1641 y Fn(D)q(if)5 b(f)516 1648 y Fk(E)566 1641 y Fr(=)13 b Fn(c)p Fr(\()p Fn(E)s Fr(\))h(=)g Fn(g)e Fm(\000)f Fr(1)g Fm(\000)g Fn(deg)r Fr(\()p Fn(E)s Fr(\))g(+)g Fn(l)q Fr(\()p Fn(E)s Fr(\))j(=)g Fn(g)e Fm(\000)f Fr(1)g Fm(\000)g Fn(deg)r Fr(\()p Fn(E)s Fr(\))0 1767 y(while)490 1836 y Fn(dim)577 1843 y Fk(k)602 1836 y Fn(D)q(if)5 b(f)714 1843 y Fk(E)763 1836 y Fm(\025)14 b Fn(dim)903
1843 y Fk(k)927 1836 y Fn(L)p Fr(\()p Fn(D)f Fm(\000)e Fn(E)s Fr(\))j(=)f Fn(l)q Fr(\()p Fn(D)g Fm(\000)e Fn(E)s Fr(\))j(=)327 1939 y(=)g Fn(deg)r Fr(\()p Fn(D)q Fr(\))e Fm(\000)f Fn(deg)r Fr(\()p Fn(E)s Fr(\))g(+)g Fn(c)p Fr(\()p Fn(D)h Fm(\000)f Fn(E)s Fr(\))j Fm(\025)g Fn(deg)r Fr(\()p Fn(D)q Fr(\))e Fm(\000)e Fn(deg)r Fr(\()p Fn(E)s Fr(\))i Fm(\000)e Fn(g)j Fr(+)e(1)0 2042 y(Mo)o(ving)k(things)h(around) f(a)i(bit)f(w)o(e)g(see)h(that)g Fn(deg)r Fr(\()p Fn(D)q Fr(\))e Fm(\024)e Fr(2\()p Fn(g)g Fm(\000)e Fr(1\).)100 2112 y(This)17 b(sho)o(ws)f(that)i(there)g(exist)g(an)g(upp)q(er)f(b)q (ound)g(on)g(the)h(p)q(ossible)f(degrees)g(of)h(divisor)f Fn(D)i Fr(for)0 2182 y(whic)o(h)i Fn(d)174 2164 y Fj(\000)p Fl(1)250 2182 y Fr(sits)h(inside)g(the)g(k)o(ernal)g(of)h(a)f(nonzero)g (W)l(eil)g(di\013eren)o(tial)f Fn(\025)p Fr(.)i(This)e(will)h(allo)o(w) g(us)g(to)0 2252 y(asso)q(ciate)16 b(a)h(\\sp)q(ecial")f(divisor)f(to)i (a)g(di\013eren)o(tial.)100 2321 y(Let)c Fn(\025)f Fr(b)q(e)h(an)o(y)f (W)l(eil)g(di\013eren)o(tial)f(not)h(equal)h(to)f(zero.)g(Then)g(there) h(exists)f(a)g(divisor)f Fn(D)j Fr(suc)o(h)e(that)0 2391 y Fn(d)26 2373 y Fj(\000)p Fl(1)93 2391 y Fm(2)i Fn(k)r(er)q Fr(\()p Fn(\025)p Fr(\).)i(W)l(e)e(examine)g(the)g(set)g(of)g(all)g (suc)o(h)f Fn(D)q Fr(.)i(\(F)l(or)e(example,)g(if)i Fn(E)h(<)e(D)h Fr(then)f Fn(E)j Fr(lies)d(in)f(this)0 2461 y(set.)i(Also,)f(if)h Fn(D)299 2468 y Fl(1)336 2461 y Fr(and)f Fn(D)472 2468 y Fl(2)510 2461 y Fr(are)g(t)o(w)o(o)g(suc)o(h)g(divisors)f(then)h Fn(sup)p Fr(\()p Fn(D)1222 2468 y Fl(1)1245 2461 y Fn(;)8 b(D)1308 2468 y Fl(2)1331 2461 y Fr(\))14 b(=)1417 2423 y Fh(P)1469 2461 y Fr(max\()p Fn(n)1611 2468 y Fl(1)1634 2461 y Fr(\()p Fn(P)7 b Fr(\))p Fn(;)h(n)1763 2468 y Fl(2)1786 2461 y Fr(\()p Fn(P)f Fr(\)\))g Fm(\001)g Fn(P)0 2531 y Fr(is)16 b(in)f(the)i(set)f(as)g(w)o(ell,)f(where)h Fn(D)630 2538 y Fk(i)661 2531 y Fr(=)713 2493 y Fh(P)774 2531 y Fn(n)804 2538 y Fk(i)821 2531 y Fr(\()p Fn(P)7 b Fr(\))k Fm(\001)f Fn(P)23 b Fr(and)15 b(the)i(sum)e(ranges)g(o)o(v)o (er)g Fn(P)21 b Fm(2)14 b Fn(C)1672 2538 y Fk(K)1710 2531 y Fr(.\))i(Hence,)h(if)0 2600 y Fn(D)h Fr(is)d(suc)o(h)g(a)h (divisor)f(of)i(largest)e(p)q(ossible)g(degree)h(\(it)g(exists)h(due)e (to)i(the)f(upp)q(er)f(b)q(ound\))h(then)g Fn(D)f Fm(\025)0 2670 y Fr(an)o(y)h(other)g(suc)o(h)g(divisor.)950 2790 y(37)p eop %%Page: 38 38 38 37 bop 100 50 a Fp(Conclusion:)13 b Fm(8)p Fn(\025)g Fm(2)h Fn(D)q(if)5 b(f)632 57 y Fk(K)675 50 y Fm(\000)s(f)p Fr(0)p Fm(g)12 b Fr(there)g(is)g(a)h(unique)e(largest)h(divisor)f Fn(D)k Fr(with)d Fn(d)1674 32 y Fj(\000)p Fl(1)1741 50 y Fm(\032)i Fn(k)r(er)q Fr(\()p Fn(\025)p Fr(\).)100 124 y Fp(Notation:)f Fr(This)g Fn(D)j Fr(is)d(denoted)g(b)o(y)g Fn(K)859 131 y Fk(\025)899 124 y Fr(or)h Fn(W)1005 131 y Fk(\025)1031 124 y Fr(,)f(and)g(is)g(called)h(the)g Fq(c)m(anonic)m(al)j(divisor)p Fr(.\))e(\(Here)0 193 y(Professor)e(Lenstra)g(p)q(oin)o(ts)h(out)g(that)h(b)q(oth)f(of)h (these)f(terrible)f(notations)h(are)g(used)f(in)h(the)h(literature)0 263 y(and)h(neither)g(of)h(them)f(are)g(his)g(fault.\))100 377 y Fp(Claim:)h Fn(W)328 384 y Fk(f)t(\025)391 377 y Fr(=)d(\()p Fn(f)5 b Fr(\))12 b(+)f Fn(W)620 384 y Fk(\025)663 377 y Fr(\()p Fn(f)20 b Fm(2)14 b Fn(K)819 359 y Fj(\003)842 377 y Fn(;)8 b(\025)14 b Fm(2)g Fn(D)q(if)5 b(f)1066 384 y Fk(K)1117 377 y Fm(\000)11 b(f)p Fr(0)p Fm(g)p Fr(\).)100 450 y(Using)16 b(the)i(fact)g(that)f Fn(L)p Fr(\()p Fn(f)5 b Fr(\))13 b Fm(\001)e Fn(D)q(if)5
b(f)785 457 y Fk(W)822 462 y Fc(\025)866 450 y Fm(\032)14 b Fn(D)q(if)5 b(f)1031 459 y Fk(W)1068 464 y Fc(\025)1094 459 y Fl(+\()p Fk(f)t Fl(\))1199 450 y Fr(\(whic)o(h)16 b(follo)o(ws)h(from)f(the)h(\014rst)g(claim)0 520 y(ab)q(o)o(v)o(e)g (with)g Fn(D)g Fr(=)f Fn(W)416 527 y Fk(\025)459 520 y Fr(and)h Fn(E)h Fr(=)d Fn(W)713 527 y Fk(\025)751 520 y Fr(+)d(\()p Fn(f)5 b Fr(\)\))19 b(w)o(e)e(see)h(that)g Fn(f)f Fm(\001)12 b Fn(\025)j Fm(2)h Fn(D)q(if)5 b(f)1444 529 y Fk(W)1481 534 y Fc(\025)1506 529 y Fl(+\()p Fk(f)t Fl(\))1612 520 y Fr(\(since)17 b Fn(f)k Fm(2)16 b Fn(L)p Fr(\()p Fn(f)5 b Fr(\))0 594 y(and)17 b Fn(\025)e Fm(2)h Fn(D)q(if)5 b(f)303 601 y Fk(W)340 606 y Fc(\025)368 594 y Fr(.\))18 b(It)g(follo)o(ws)e(that)i(\()p Fn(W)813 601 y Fk(\025)851 594 y Fr(+)12 b(\()p Fn(f)5 b Fr(\)\))988 576 y Fj(\000)p Fl(1)1058 594 y Fm(\032)15 b Fn(k)r(er)q Fr(\()p Fn(f)5 b(\025)p Fr(\))20 b(and)d(th)o(us)f(\()p Fn(f)5 b Fr(\))13 b(+)f Fn(W)1688 601 y Fk(\025)1729 594 y Fm(\024)j Fn(W)1830 601 y Fk(f)t(\025)1897 594 y Fr(b)o(y)0 663 y(the)i(de\014nition)e(of)i Fn(W)411 670 y Fk(f)t(\025)460 663 y Fr(.)g(The)f Fm(\025)h Fr(case)f(follo)o (ws)g(b)o(y)g(symmetry)l(.)100 777 y(No)o(w)c(supp)q(ose)f(that)h Fn(\025)p Fr(,)g Fn(\026)h Fr(are)e(linearly)h(indep)q(enden)o(t)f(in)g Fn(D)q(if)5 b(f)1289 784 y Fk(K)1330 777 y Fr(.)12 b(W)l(e)g(lo)q(ok)h (for)e(a)i(con)o(tradiction:)100 851 y(As)j(ab)q(o)o(v)o(e)g(w)o(e)h (ha)o(v)o(e)712 932 y Fn(L)p Fr(\()p Fn(W)812 939 y Fk(\025)850 932 y Fm(\000)11 b Fn(E)s Fr(\))g Fm(\001)g Fn(\025)j Fm(\032)f Fn(D)q(if)5 b(f)1202 939 y Fk(E)0 1049 y Fr(Th)o(us,)525 1131 y Fn(L)p Fr(\()p Fn(W)625 1138 y Fk(\025)663 1131 y Fm(\000)11 b Fn(E)s Fr(\))g Fm(\001)g Fn(\025)g Fr(+)g Fn(L)p Fr(\()p Fn(W)998 1138 y Fk(\026)1036 1131 y Fm(\000)g Fn(E)s Fr(\))g Fm(\001)g Fn(\026)i Fm(\032)h Fn(D)q(if)5 b(f)1389 1138 y Fk(E)0 1248 y Fr(Since)16 b Fn(\025)h Fr(and)f Fn(\026)g Fr(are)g(indep)q(enden)o(t)g(w)o(e)g(ha)o(v)o(e)g (the)h(inequalit)o(y)f(of)g(dimensions)623 1389 y Fn(l)q Fr(\()p Fn(W)705 1396 y Fk(\025)743 1389 y Fm(\000)10 b Fn(E)s Fr(\))h(+)g Fn(l)q Fr(\(\()p Fn(W)1013 1396 y Fk(\026)1052 1389 y Fm(\000)g Fn(E)s Fr(\))i Fm(\024)h Fn(c)p Fr(\()p Fn(E)s Fr(\))100 1535 y(No)o(w)g(let)h Fn(deg)r Fr(\()p Fn(E)s Fr(\))f Fm(!)g(\0001)p Fr(.)g(Using)g (Riemann's)e(Theorem)i(and)g(the)h(fact)g(that)g Fn(deg)r Fr(\()p Fn(D)q Fr(\))g Fn(>)e Fr(2)p Fn(g)c Fm(\000)e Fr(2)0 1604 y(implies)15 b Fn(c)p Fr(\()p Fn(D)q Fr(\))g(=)e(0)k(w)o(e) f(ha)o(v)o(e)g(that)598 1746 y Fn(deg)r Fr(\()p Fn(W)739 1753 y Fk(\025)776 1746 y Fm(\000)11 b Fn(E)s Fr(\))g Fm(\000)g Fn(l)q Fr(\()p Fn(W)1028 1753 y Fk(\025)1065 1746 y Fm(\000)g Fn(E)s Fr(\))j(=)g Fn(g)e Fm(\000)f Fr(1)0 1888 y(i.e.)16 b Fn(deg)r Fr(\()p Fm(\000)p Fn(E)s Fr(\))d(=)h Fn(l)q Fr(\()p Fn(W)420 1895 y Fk(\025)457 1888 y Fm(\000)9 b Fn(E)s Fr(\))17 b(plus)d(some)i(constan)o(t.)f (Similarly)f Fn(deg)r Fr(\()p Fm(\000)p Fn(E)s Fr(\))g(=)g Fn(l)q Fr(\()p Fn(W)1576 1895 y Fk(\026)1613 1888 y Fm(\000)c Fn(E)s Fr(\))16 b(plus)f(some)0 1957 y(constan)o(t.)22 b(But)i(Riemann's)c(Theorem)i(applied)g(to)h(the)h(divisor)d(E)i(alone) g(implies)e(that)j Fn(c)p Fr(\()p Fn(E)s Fr(\))g(=)0 2027 y Fm(\000)p Fn(deg)r Fr(\()p Fn(E)s Fr(\))11 b(+)g(constan)o(t)o (.)16 b(W)l(e)h(are)f(forced)h(to)f(conclude)g(that)578 2183 y(2)11 b Fm(\001)f Fn(deg)r Fr(\()p Fm(\000)p Fn(E)s Fr(\))k Fm(\024)g(\000)p Fn(deg)r Fr(\()p Fn(E)s Fr(\))d(+)f(\(constan) o(t\))0 2299 y(as)16 b Fn(deg)r Fr(\()p Fn(E)s Fr(\))h(go)q(es)f(to)h
(negativ)o(e)g(in\014nit)o(y)l(.)e(This)h(is)g(a)g(con)o(tradiction.) 100 2373 y(=)-8 b Fm(\))15 b Fn(\025)i Fr(and)f Fn(\026)h Fr(are)f(dep)q(enden)o(t.)100 2447 y(=)-8 b Fm(\))15 b Fn(dim)283 2454 y Fk(K)322 2447 y Fn(D)q(if)5 b(f)434 2454 y Fk(K)488 2447 y Fr(=)13 b(1.)0 2596 y Fp(Lecture)20 b(18)c Fr(\(Octob)q(er)h(9,)f(1995\).)0 2670 y(Scrib)q(e:)g(Nghi)g (Nguy)o(en)950 2790 y(38)p eop %%Page: 39 39 39 38 bop 100 50 a Fr(W)l(e)25 b(recall)f(the)h(notations:)f Fn(k)29 b Fm(\032)e Fn(K)i Fr(is)24 b(a)h(function)f(\014eld,)g Fn(D)j Fr(is)d(an)h(arbitrary)e(divisor)g(on)0 126 y(the)f(curv)o(e)f Fn(C)265 133 y Fk(K)303 126 y Fr(,)g Fn(d)p 338 134 26 2 v 22 w Fr(=)447 89 y Fh(Q)511 126 y Fr(^)-34 b Fn(m)546 101 y Fk(n)p Fl(\()p Fk(P)5 b Fl(\))546 132 y Fk(p)635 126 y Fr(,)21 b Fn(l)q Fr(\()p Fn(D)q Fr(\))i(=)f(dim)933 133 y Fk(k)966 126 y Fn(L)p Fr(\()p Fn(D)q Fr(\),)g Fn(L)p Fr(\()p Fn(D)q Fr(\))i(=)d Fn(K)e Fm(\\)14 b Fn(d)p 1422 134 V -22 x Fj(\000)p Fl(1)1523 126 y Fr(inside)20 b Fp(A)1712 133 y Fk(K)1751 126 y Fr(,)h Fn(c)p Fr(\()p Fn(D)q Fr(\))i(=)0 199 y(dim)83 206 y Fk(k)116 199 y Fp(A)159 206 y Fk(k)184 199 y Fn(=)p Fr(\()p Fn(K)15 b Fr(+)c Fn(d)p 335 207 V -23 x Fj(\000)p Fl(1)414 199 y Fr(\),)658 316 y Fn(D)q(if)5 b(f)770 323 y Fk(K)824 316 y Fr(=)14 b Fm(f)p Fr(W)l(eil)i(di\013eren)o(tials)n Fm(g)180 403 y Fr(=)e Fm(f)p Fn(k)e Fm(\000)f Fr(linear)16 b(maps)f Fn(\025)f Fr(:)f Fp(A)728 410 y Fk(K)781 403 y Fm(\000)-9 b(!)14 b Fn(k)k Fr(whic)o(h)e(are)g(zero)g(on)g(K)h(and)f (on)g(some)g Fn(d)p 1666 411 V 1691 381 a Fj(\000)p Fl(1)1745 403 y Fm(g)294 498 y Fr(=)347 450 y Fh([)357 557 y Fk(D)402 498 y Fr(\()p Fn(D)q(if)5 b(f)533 505 y Fk(D)571 498 y Fr(\))17 b(where)f Fn(D)q(if)5 b(f)863 505 y Fk(D)915 498 y Fr(=)14 b Fm(f)p Fn(\025)g Fm(2)30 b Fn(D)q(if)5 b(f)1211 505 y Fk(K)1268 498 y Fr(with)17 b Fn(d)p 1382 506 V -23 x Fj(\000)p Fl(1)1475 498 y Fm(\032)d Fn(k)r(er)q(\025)p Fm(g)0 677 y Fr(,)23 b(so)g Fn(c)p Fr(\()p Fn(D)q Fr(\))i(=)g(dim)379 684 y Fk(k)412 677 y Fn(D)q(if)5 b(f)524 684 y Fk(D)562 677 y Fr(.)23 b(F)l(or)f(eac)o(h)h(0)g Fm(6)p Fr(=)h Fn(\025)h Fm(2)g Fn(D)q(if)5 b(f)1149 684 y Fk(K)1213 677 y Fr(there)23 b(is)f(a)h(unique)g(largest)f Fn(D)j Fr(with)0 746 y Fn(\025)16 b Fm(2)h Fn(D)q(if)5 b(f)207 753 y Fk(K)247 746 y Fr(;)18 b(this)g Fn(D)i Fr(is)d(denoted)h(b)o(y)g Fn(K)793 753 y Fk(D)847 746 y Fr(or)g Fn(W)957 753 y Fk(\025)1001 746 y Fr(\("canonical)g(divisor"\).)f(Last)h(time)g(w)o(e) g(pro)o(v)o(ed:)0 816 y(dim)83 823 y Fk(K)130 816 y Fn(D)q(if)5 b(f)242 823 y Fk(K)308 816 y Fr(=)26 b(1.)d(Also,)h Fn(W)616 823 y Fk(f)t(\025)691 816 y Fr(=)i(\()p Fn(f)5 b Fr(\))17 b(+)f Fn(W)942 823 y Fk(\025)992 816 y Fr(for)24 b Fn(f)31 b Fm(2)c Fn(K)1237 798 y Fj(\003)1259 816 y Fr(.)d(So)f(the)i(divisor)d (class)h(of)h Fn(K)1866 823 y Fk(\025)1917 816 y Fr(is)0 886 y(indep)q(enden)o(t)15 b(of)i Fn(\025)p Fr(.)100 996 y(No)o(w)h(pic)o(k)h Fn(\025)e Fm(2)h Fn(D)q(if)5 b(f)532 1003 y Fk(K)573 996 y Fr(,)18 b Fm(6)p Fr(=)g(0.)g(Then)g Fn(D)q(if)5 b(f)963 1003 y Fk(K)1021 996 y Fr(=)18 b Fn(K)q(:\025)p Fr(.)g(If)i Fn(f)j Fm(2)18 b Fn(K)1393 978 y Fj(\003)1415 996 y Fr(,)h(then)g Fn(f)5 b(\025)18 b Fm(2)g Fn(D)q(if)5 b(f)1803 1003 y Fk(D)1859 996 y Fm(\()-8 b(\))0 1066 y Fn(W)47 1073 y Fk(f)t(\025)110 1066 y Fm(\025)14 b Fn(D)h Fm(\()-8 b(\))14 b Fr(\()p Fn(f)5 b Fr(\))12 b(+)f Fn(W)501 1073 y Fk(\025)541 1066
y Fm(\025)i Fn(D)j Fm(\()-8 b(\))13 b Fn(f)20 b Fm(2)14 b Fn(L)p Fr(\()p Fn(W)946 1073 y Fk(\025)983 1066 y Fm(\000)d Fn(D)q Fr(\).)18 b(This)d(argumen)o(t)g(giv)o(es)100 1176 y Fp(Riemann-Ro)r(c)n(h)j(Theorem:)602 1308 y Fn(l)q Fr(\()p Fn(D)q Fr(\))13 b Fm(\000)d Fn(l)q Fr(\()p Fn(W)842 1315 y Fk(\025)880 1308 y Fm(\000)h Fn(D)q Fr(\))k(=)e Fn(deg)r(D)f Fr(+)f(1)g Fm(\000)g Fn(g)100 1481 y Fp(Remarks:)100 1552 y Fn(l)q Fr(\()p Fn(D)q Fr(\))k(=)e(0)k(if)j Fn(deg)r(D)15 b(<)f Fr(0,)100 1622 y Fn(l)q Fr(\()p Fn(W)182 1629 y Fk(\025)208 1622 y Fr(\))g(=)g Fn(g)r Fr(,)100 1693 y Fn(deg)r Fr(\()p Fn(W)241 1700 y Fk(\025)267 1693 y Fr(\))g(=)g(2)p Fn(g)e Fm(\000)f Fr(2,)100 1763 y(and)k Fn(l)q Fr(\()p Fn(D)q Fr(\))h(=)d Fn(deg)r(D)f Fr(+)f(1)g Fm(\000)g Fn(g)18 b Fr(if)j Fn(deg)r(D)15 b(>)e Fr(2)p Fn(g)g Fm(\000)e Fr(2.)0 1897 y Fp(Lecture)20 b(19)c Fr(\(Octob)q(er)h(11,)f(1995\).)0 1968 y(Scrib)q(e:)g(Da)o(vid)g(M.)g(Jones)0 2101 y Fo(Recall)p Fr(:)g Fn(`)p Fr(\()p Fn(D)q Fr(\))c Fm(\000)f Fn(`)p Fr(\()p Fn(W)414 2108 y Fk(\025)451 2101 y Fm(\000)g Fn(D)q Fr(\))k(=)f(deg)8 b Fn(D)13 b Fr(+)e(1)g Fm(\000)g Fn(g)170 2172 y(`)p Fr(\()p Fn(D)q Fr(\))k(=)f(0)j(if)f(deg)9 b Fn(D)15 b(<)f Fr(0.)170 2242 y Fn(`)p Fr(\()p Fn(D)q Fr(\))h(=)f(deg)9 b Fn(D)j Fr(+)f(1)g Fm(\000)g Fn(g)18 b Fr(if)f(deg)9 b Fn(D)15 b(>)f Fr(2)p Fn(g)e Fm(\000)f Fr(2.)170 2313 y Fn(K)18 b Fr(=)c Fn(k)r Fr(\()p Fn(t)p Fr(\))j(has)e Fn(g)h Fr(=)d(0.)0 2447 y Fo(Recall)j(from)g(the)h(pro)q (of)f(of)h(the)f(Riemann-Ro)q(c)o(h)e(Theorem)p Fr(:)100 2529 y Fn(k)r Fr(\()p Fn(t)p Fr(\))186 2492 y Fk(m)184 2529 y Fm(\032)p Fn(K)t Fr(,)i(\()p Fn(t)p Fr(\))e(=)g Fn(N)j Fm(\000)10 b Fn(D)19 b Fr(where)d Fn(N)s(;)8 b(D)18 b Fr(are)e(e\013ectiv)o(e,)h(coprime)e(divisors.)100 2599 y(If)h Fn(K)i Fr(=)262 2562 y Fh(L)317 2574 y Fk(m)317 2614 y(i)p Fl(=1)392 2599 y Fn(k)r Fr(\()p Fn(t)p Fr(\))12 b Fm(\001)f Fn(u)542 2606 y Fk(i)574 2599 y Fr(and)16 b(\()p Fn(u)719 2606 y Fk(i)736 2599 y Fr(\))e Fm(\025)f(\000)p Fr(\()p Fn(a)905 2606 y Fk(i)934 2599 y Fr(+)d(1\))i Fm(\001)f Fn(D)q Fr(,)17 b(then)f Fn(g)g Fm(\024)d Fr(1)e(+)1428 2562 y Fh(P)1480 2574 y Fk(m)1480 2614 y(i)p Fl(=1)1556 2599 y Fn(a)1582 2606 y Fk(i)1599 2599 y Fr(.)100 2670 y(In)16 b(the)h(case)f Fn(K)i Fr(=)13 b Fn(k)r Fr(\()p Fn(t)p Fr(\),)k Fn(m)d Fr(=)f(1)p Fn(;)8 b(u)766 2677 y Fl(1)802 2670 y Fr(=)14 b(1)p Fn(;)8 b(a)928 2677 y Fl(1)964 2670 y Fr(=)14 b Fm(\000)p Fr(1)i(so)g Fn(g)f Fm(\024)f Fr(0)g Fm(\))f Fn(g)j Fr(=)d(0.)950 2790 y(39)p eop %%Page: 40 40 40 39 bop 0 50 a Fr(Another)20 b(pro)q(of)g(that)g Fn(g)h Fr(=)f(0)g(for)g Fn(K)j Fr(=)d Fn(k)r Fr(\()p Fn(t)p Fr(\):)g Fn(g)h Fr(=)f(dim)1102 57 y Fk(k)1135 50 y Fp(A)1178 57 y Fk(K)1216 50 y Fn(=)p Fr(\()p Fn(K)e Fr(+)1386 37 y(^)1372 50 y Fm(O)q Fr(\).)i(But)1582 37 y(^)1569 50 y Fm(O)h Fr(=)1688 12 y Fh(Q)1735 65 y Fk(P)1790 37 y Fr(^)1777 50 y Fm(O)1817 57 y Fk(P)1870 50 y Fr(and)0 123 y Fn(K)15 b Fr(+)120 110 y(^)107 123 y Fm(O)g Fr(=)e Fp(A)257 130 y Fk(K)313 123 y Fr(so)j Fn(g)f Fr(=)f(dim)549 130 y Fk(k)573 123 y Fm(f)p Fr(0)p Fm(g)g Fr(=)f(0.)0 237 y(Since)j Fn(g)f Fr(=)f(0,)i(w)o(e)h(ha)o(v)o(e:)e Fn(`)p Fr(\()p Fn(D)q Fr(\))g(=)f(0)i(if)h(deg)9 b Fn(D)15 b(<)f Fr(0)i(and)482 307 y Fn(`)p Fr(\()p Fn(D)q Fr(\))f(=)f(deg)9 b Fn(D)j Fr(+)f(1)17 b(if)f(deg)9 b Fn(D)15 b(>)f Fm(\000)p
Fr(2.)0 448 y(W)l(e)j(ha)o(v)o(e)f Fn(K)258 417 y Fk(')247 448 y Fm(!)o Fp(A)339 455 y Fk(K)378 448 y Fn(=)p Fr(\()422 410 y Fh(Q)469 462 y Fk(P)520 448 y Fr(^)-34 b Fn(m)p 511 456 44 2 v -26 x Fk(n)p Fl(\()p Fk(P)5 b Fl(\))555 462 y Fk(P)644 448 y Fr(\),)16 b(where)837 377 y Fh(\032)883 418 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))14 b(=)g(0)155 b Fm(8)p Fn(P)19 b Fm(6)p Fr(=)14 b Fm(1)883 477 y Fn(n)p Fr(\()p Fm(1)p Fr(\))g(=)g(0)p Fn(;)8 b Fr(1)p Fn(;)g Fr(2)100 553 y Fm(1)16 b Fr(is)h(the)g(p)q(oin)o(t)f(in)505 541 y(^)491 553 y Fm(O)i Fr(for)f(whic)o(h)e Fm(O)808 560 y Fj(1)864 553 y Fr(=)f Fn(k)r Fr([)p Fn(t)977 535 y Fj(\000)p Fl(1)1030 553 y Fr(])1044 563 y Fl(\()p Fk(t)1075 553 y Fi(\000)p Fd(1)1121 563 y Fl(\))1153 553 y Fr(=)g Fm(f)1237 531 y Fk(f)p 1237 542 24 2 v 1238 570 a(g)1281 553 y Fm(2)g Fn(K)k Fr(:)c Fn(f)s(;)8 b(g)16 b Fm(2)f Fn(k)r Fr([)p Fn(t)p Fr(])p Fn(;)8 b(g)15 b Fm(6)p Fr(=)f(0)p Fn(;)8 b Fr(deg)g Fn(g)16 b Fm(\025)0 626 y Fr(deg)9 b Fn(f)c Fm(g)p Fr(.)17 b(As)f Fn(n)p Fr(\()p Fm(1)p Fr(\))f(=)e(0)p Fn(;)8 b Fr(1)p Fn(;)g Fr(2,)17 b(w)o(e)f(get)h(dim)o (\(cok)o(er\()p Fn(')p Fr(\)\))e(=)e(0)p Fn(;)8 b Fr(0)p Fn(;)g Fr(1.)0 740 y Fo(Question)p Fr(:)15 b(Ho)o(w)i(do)q(es)f(this)g (relate)h(to)g(Riemann-Ro)q(c)o(h?)100 810 y Fo(Answ)o(er)p Fr(:)e Fn(c)p Fr(\()p Fn(D)q Fr(\))g(=)e Fn(`)p Fr(\()p Fn(W)547 817 y Fk(\025)585 810 y Fm(\000)e Fn(D)q Fr(\))j(=)g(dim)846 817 y Fk(k)879 810 y Fp(A)922 817 y Fk(K)960 810 y Fn(=)p Fr(\()p Fn(K)i Fr(+)10 b Fn(d)p 1111 818 26 2 v -23 x Fj(\000)p Fl(1)1191 810 y Fr(\).)17 b(T)l(ak)o(e)f Fn(D)g Fr(=)d(0)p Fn(;)8 b Fm(\0001)p Fn(;)g Fm(\000)p Fr(2)j Fm(\001)g(1)p Fr(.)0 924 y Fo(No)o(w)16 b(w)o(e)h(w)o(ould)e(lik)o(e)h (to)h(\014nd)f(a)g(basis)g(for)g Fn(K)i Fr(=)13 b Fn(k)r Fr(\()p Fn(t)p Fr(\))k Fo(o)o(v)o(er)f Fn(k)r Fr(:)100 994 y("partial)f(fraction)h(expansion":)100 1017 y Fh(\()148 1063 y Fn(t)166 1045 y Fk(i)271 1063 y Fr(s.t.)h Fn(i)d Fm(2)g Fp(Z)468 1070 y Fj(\025)p Fl(0)163 1113 y Fk(t)178 1098 y Fc(i)p 154 1122 48 2 v 154 1150 a Fk(f)177 1140 y Fc(n)208 1133 y Fn(;)49 b(f)20 b Fm(2)14 b Fn(k)r Fr([)p Fn(t)p Fr(])i(monic,)f(irred.;)g Fn(n)f Fm(2)g Fp(Z)885 1140 y Fk(>)p Fl(0)939 1133 y Fn(;)8 b Fr(0)13 b Fm(\024)h Fn(i)g Fm(\024)g Fr(deg)8 b Fn(f)s(;)g(i)14 b Fm(2)g Fp(Z)100 1251 y Fo(F)l(or)h(no)o(w,)h(assume)f(that)i Fn(k)e Fr(=)682 1238 y(\026)681 1251 y Fn(k)q Fr(:)739 1181 y Fh(\032)784 1216 y Fn(t)802 1198 y Fk(i)974 1216 y Fn(i)f Fm(2)g Fp(Z)1087 1223 y Fj(\025)p Fl(0)844 1256 y(1)p 790 1265 128 2 v 790 1293 a(\()p Fk(t)p Fj(\000)p Fk(\013)p Fl(\))894 1283 y Fc(n)974 1276 y Fn(\013)g Fm(2)g Fn(k)r(;)8 b(n)13 b Fm(2)h Fp(Z)1242 1283 y Fk(>)p Fl(0)100 1365 y Fn(K)157 1335 y Fk(')146 1365 y Fm(!)o Fp(A)238 1372 y Fk(K)277 1365 y Fn(=)310 1328 y Fh(Q)357 1380 y Fk(P)408 1365 y Fr(^)-34 b Fn(m)p 399 1373 44 2 v -26 x Fk(n)p Fl(\()p Fk(P)5 b Fl(\))443 1380 y Fk(P)545 1365 y Fr(=)598 1328 y Fh(L)653 1380 y Fk(P)686 1365 y Fr(\()p Fn(K)747 1372 y Fk(P)781 1365 y Fn(=)10 b Fr(^)-35 b Fn(m)p 806 1373 V -26 x Fk(n)p Fl(\()p Fk(P)5 b Fl(\))850 1380 y Fk(P)939 1365 y Fr(\).)100 1443 y(No)o(w,)16 b Fp(A)270 1450 y Fk(K)309 1443 y Fn(=)342 1406 y Fh(Q)389 1458 y Fk(P)440 1443 y Fr(^)-35 b Fn(m)p 430 1451 V -26 x Fk(n)p Fl(\()p Fk(P)5 b Fl(\))474 1458 y Fk(P)577 1443 y Fr(=)643 1431 y(^)630 1443 y Fm(O)17 b Fr(if)g(all)f Fn(n)p Fr(\()p Fn(P)7 b Fr(\))14 b(=)g(0.)100 1513 y
Fm(f)i Fr(closed)g(p)q(oin)o(ts)g Fn(P)7 b Fm(g)13 b($)h Fn(k)f Fm([)e(f1g)16 b Fr(via)h Fn(k)r Fr([)p Fn(t)p Fr(])934 1522 y Fl(\()p Fk(t)p Fj(\000)p Fk(\013)p Fl(\))1053 1513 y Fm($)c Fn(\013)100 1589 y Fr(1.)24 b(If)h Fn(K)263 1596 y Fk(P)323 1589 y Fr(=)i Fn(k)r Fr(\(\()p Fn(t)17 b Fm(\000)f Fn(\013)p Fr(\)\))25 b(and)758 1576 y(^)745 1589 y Fm(O)785 1596 y Fk(P)845 1589 y Fr(=)h Fn(k)r Fr([[)p Fn(t)16 b Fm(\000)g Fn(\013)p Fr(]],)24 b(then)g(a)h(basis)e (for)h Fn(K)1580 1596 y Fk(P)1614 1589 y Fn(=)1652 1576 y Fr(^)1639 1589 y Fm(O)1679 1596 y Fk(P)1736 1589 y Fr(o)o(v)o(er)g Fn(k)i Fr(is:)60 1639 y Fl(1)p 6 1647 128 2 v 6 1676 a(\()p Fk(t)p Fj(\000)p Fk(\013)p Fl(\))110 1666 y Fc(n)140 1659 y Fn(;)8 b(n)13 b Fm(2)i Fp(Z)288 1666 y Fk(>)p Fl(0)341 1659 y Fr(.)100 1743 y(2.)h(If)h Fn(P)22 b Fr(=)14 b Fm(1)p Fr(,)i(then)h Fn(K)548 1750 y Fj(1)605 1743 y Fr(=)d Fn(k)r Fr(\(\()p Fn(t)742 1725 y Fj(\000)p Fl(1)796 1743 y Fr(\)\))h Fm(\033)915 1730 y Fr(^)902 1743 y Fm(O)942 1750 y Fj(1)998 1743 y Fr(=)f Fn(k)r Fr([[)p Fn(t)1125 1725 y Fj(\000)p Fl(1)1178 1743 y Fr(]].)i(Then)g(a)h(basis)f(for)g Fn(K)1649 1750 y Fj(1)1692 1743 y Fn(=)1730 1730 y Fr(^)1717 1743 y Fm(O)1757 1750 y Fj(1)1816 1743 y Fr(o)o(v)o(er)f Fn(k)0 1813 y Fr(is:)h Fn(t)82 1795 y Fk(i)98 1813 y Fn(;)8 b(i)15 b Fm(2)f Fp(Z)234 1820 y Fk(>)p Fl(0)287 1813 y Fr(.)j(A)g(basis)e(for) h Fn(K)612 1820 y Fj(1)655 1813 y Fn(=)9 b Fr(^)-34 b Fn(m)p 680 1821 44 2 v 723 1790 a Fk(n)723 1825 y Fj(1)782 1813 y Fr(o)o(v)o(er)16 b Fn(k)i Fr(is:)e Fn(t)1015 1795 y Fk(i)1031 1813 y Fn(;)8 b(i)15 b Fm(2)f Fp(Z)1167 1820 y Fj(\025)p Fl(1)p Fj(\000)p Fk(n)1292 1813 y Fr(\(esp.)j Fn(n)c Fr(=)h(1)p Fn(;)8 b Fr(2\).)100 1883 y(Supp)q(ose)14 b Fn(n)p Fr(\()p Fm(1)p Fr(\))h(=)e(0.)j(Then)f Fn(')h Fr(is)f(on)o(to)g(with)h(k)o(ernel)f(=)f Fn(k)r Fr(.)h(\(Ev)o(ery)h (basis)f(elemen)o(t)g(of)h Fn(k)r Fr(\()p Fn(t)p Fr(\))p Fn(=k)i Fr(is)0 1958 y(sen)o(t)e(to)h(itself)g(in)344 1921 y Fh(L)399 1973 y Fk(P)441 1958 y Fn(K)483 1965 y Fk(P)516 1958 y Fn(=)9 b Fr(^)-34 b Fn(m)p 541 1966 V -26 x Fk(n)p Fl(\()p Fk(P)5 b Fl(\))585 1973 y Fk(P)690 1958 y Fr(except)18 b(1)e(whic)o(h)g(is)g(sen)o(t)g(to)h(0.\))100 2031 y(Supp)q(ose)f Fn(n)p Fr(\()p Fm(1)p Fr(\))h(=)e(1.)j(Then)f Fn(')h Fr(is)f(bijectiv)o(e,)h Fp(A)1031 2038 y Fk(K)1086 2031 y Fr(=)d Fn(K)h Fm(\010)c Fr(\()e(^)-35 b Fn(m)p 1268 2039 V 13 x Fj(1)1366 2031 y Fm(\010)1417 1994 y Fh(Q)1464 2046 y Fk(P)5 b Fj(6)p Fl(=)p Fj(1)1589 2019 y Fr(^)1576 2031 y Fm(O)1616 2038 y Fk(P)1649 2031 y Fr(\))18 b(so)f Fn(D)h Fr(=)d Fm(\0001)0 2102 y Fr(has)h Fn(`)p Fr(\()p Fn(D)q Fr(\))f(=)e(0)p Fn(;)8 b(c)p Fr(\()p Fn(D)q Fr(\))15 b(=)f(0)g(=)f Fn(`)p Fr(\()p Fn(W)652 2109 y Fk(\025)689 2102 y Fm(\000)e Fn(D)q Fr(\).)100 2171 y(Supp)q(ose)k Fn(n)p Fr(\()p Fm(1)p Fr(\))g(=)e(2.)k(Then)f Fn(')g Fr(is)g(injectiv)o(e)h(and)f(has)g(cok)o(ernel)g(of)h(dimension) e(1.)h Fn(D)g Fr(=)d Fm(\000)p Fr(2)e Fm(\001)g(1)0 2241 y Fr(has)18 b Fn(`)p Fr(\()p Fn(D)q Fr(\))i(=)e(0)p Fn(;)8 b(c)p Fr(\()p Fn(D)q Fr(\))19 b(=)f(1)g(=)g Fn(`)p Fr(\()p Fn(W)681 2248 y Fk(\025)720 2241 y Fm(\000)13 b Fn(D)q Fr(\).)20 b(Also,)f Fn(c)p Fr(\()p Fn(D)q Fr(\))g(=)f(1)g(=)g(dim)1357 2248 y Fk(k)1390 2241 y Fr(Di\013)1475 2248 y Fk(D)1511 2241 y Fr(.)h(Di\013)1629 2248 y Fk(D)1683 2241 y Fr(=)g(the)g(set)h (of)0 2312 y Fn(k)r Fr(-linear)15 b(functions)g Fp(A)440 2319 y Fk(K)479 2312 y Fn(=)p Fr(\()10 b(^)-35 b Fn(m)p
523 2320 V -22 x Fl(2)567 2324 y Fj(1)620 2312 y Fm(\010)670 2275 y Fh(L)726 2327 y Fk(P)5 b Fj(6)p Fl(=)p Fj(1)851 2299 y Fr(^)837 2312 y Fm(O)877 2319 y Fk(P)910 2312 y Fr(\))14 b Fm(!)g Fn(k)k Fr(that)f(are)f(0)h(on)f(the)h(image)f(of)g Fn(K)t Fr(.)100 2382 y(\(An)k(example)g(of)h(suc)o(h)e(a)h(function)g Fn(\025)g Fm(6)p Fr(=)g(0)g(is)g(giv)o(en)g(as)f(follo)o(ws:)h(T)l(ak)o (e)g Fn(x)g Fm(2)h Fp(A)1680 2389 y Fk(K)1739 2382 y Fr(and)e(write)0 2453 y Fn(x)d Fr(=)f Fn(y)e Fr(+)f Fn(z)19 b Fr(where)e Fn(y)h Fm(2)d Fn(K)21 b Fr(and)c Fn(z)h Fm(2)d Fr(\()10 b(^)-35 b Fn(m)p 732 2461 V 12 x Fj(1)830 2453 y Fm(\010)880 2416 y Fh(Q)927 2468 y Fk(P)5 b Fj(6)p Fl(=)p Fj(1)1053 2440 y Fr(^)1039 2453 y Fm(O)1079 2460 y Fk(P)1112 2453 y Fr(\),)18 b(where)e Fn(z)1330 2460 y Fj(1)1388 2453 y Fr(=)h(\(comp)q(onen)o(t)g(of)g Fn(z)j Fr(at)e Fm(1)p Fr(\))0 2531 y Fm(\021)c Fn(c)c Fm(\001)h Fn(t)128 2512 y Fj(\000)p Fl(1)198 2531 y Fr(mo)q(d)26 b(^)-35 b Fn(m)p 310 2538 V -23 x Fl(2)354 2543 y Fj(1)413 2531 y Fr(and)16 b Fn(c)d Fm(2)h Fn(k)620 2512 y Fj(\003)643 2531 y Fr(.\))100 2600 y(That's)e(it)g(for)g(the)h(case)g Fn(k)i Fr(=)650 2587 y(\026)648 2600 y Fn(k)r Fr(.)d(As)h(an)f (exercise,)g(w)o(ork)g(out)g(this)g(basis)g(in)g(the)g (non-algebraically)0 2670 y(closed)k(case.)950 2790 y(40)p eop %%Page: 41 41 41 40 bop 0 50 a Fr(Ab)q(o)o(v)o(e,)21 b Fn(\025)g Fr(is)f(0)h(on)g Fn(d)p 396 58 26 2 v -22 x Fj(\000)p Fl(1)496 50 y Fr(where)g Fn(d)p 645 58 V 21 w Fm($)g Fn(D)h Fr(=)f Fm(\000)p Fr(2)14 b Fm(\001)g(1)p Fr(.)20 b(Actually)l(,)h Fn(D)i Fr(is)d(the)i(largest)e Fn(D)1708 32 y Fj(0)1743 50 y Fr(for)h(whic)o(h)0 120 y Fn(\025)14 b Fm(2)g Fr(Di\013)175 127 y Fk(D)209 117 y Fi(0)224 120 y Fr(.)j(So)f Fn(W)371 127 y Fk(\025)411 120 y Fr(=)e Fm(\000)p Fr(2)c Fm(\001)h(1)p Fr(.)100 189 y Fn(d)p 100 197 V -22 x Fj(\000)p Fl(1)193 189 y Fr(=)i Fm(f)p Fn(z)k Fm(2)d Fp(A)400 196 y Fk(K)452 189 y Fr(:)g Fn(\025)p Fr(\()542 177 y(^)528 189 y Fm(O)e(\001)f Fn(z)r Fr(\))k(=)e(0)p Fm(g)p Fr(.)0 307 y Fo(Remark)p Fr(:)19 b(One)h(could)g(carry)g(out)g(these)h(calculations)e(replacing) g Fp(A)1343 314 y Fk(K)1403 307 y Fr(with)h(the)h("non-complete")0 375 y(analogue)211 338 y Fh(Q)258 350 y Fj(0)258 390 y Fk(P)300 375 y Fn(K)t Fr(,)f(restricted)g(with)g(resp)q(ect)g(to)962 338 y Fh(Q)1009 390 y Fk(P)1064 363 y Fr(^)1050 375 y Fm(O)1090 382 y Fk(P)1123 375 y Fr(.)g(Ho)o(w)o(ev)o(er,)g(adele)g (rings)e(ha)o(v)o(e)i(v)o(ery)g(nice)0 445 y("structural")h(prop)q (erties.)h(F)l(or)f(example,)i(if)f Fn(L=K)27 b Fr(is)22 b(complete)h(then)g Fp(A)1467 452 y Fk(K)1521 445 y Fm(\012)1560 452 y Fk(K)1613 445 y Fn(L)h Fm(!)g Fp(A)1788 452 y Fk(L)1841 445 y Fr(is)e(an)0 515 y(isomorphism.)13 b(This)j(is)g(not)g(true)h(if) f(y)o(ou)h(are)f(not)g(using)g(complete)g(\014elds.)0 632 y Fo(F)-6 b(A)o(CT)p Fr(:)16 b(If)h Fn(g)f Fr(=)d(0,)j(then)h Fn(K)536 619 y Fm(\030)536 634 y Fr(=)574 639 y Fk(k)613 632 y Fn(k)r Fr(\()p Fn(t)p Fr(\))d Fm(,)f(9)k Fr(closed)f Fn(P)k Fr(:)14 b(deg)9 b Fn(P)21 b Fr(=)13 b(1.)100 702 y(Pro)q(of:)j Fm(\))e Fn(P)20 b Fr(=)14 b Fm(1)i Fr(has)g(degree)g(1.) 256 771 y Fm(\()f Fr(deg)9 b Fn(P)22 b Fr(=)15 b(1.)j(Then)f Fn(`)p Fr(\()p Fn(P)7 b Fr(\))16 b(=)f(2.)j(So)f Fn(L)p Fr(\()p Fn(P)7 b Fr(\))16 b Fm(\033)f Fn(k)k Fr(but)f Fn(L)p Fr(\()p Fn(P)7 b Fr(\))16 b Fm(6)p Fr(=)f Fn(k)k Fr(so)e Fm(9)p Fn(x)f Fm(2)g Fn(L)p Fr(\()p Fn(P)7 b Fr(\))18 b(with)0 841 y Fn(x)c Fm(62)g Fn(k)r Fr(.)i(Then)g(the)h(p)q
(ole)g(divisor)e(of)i Fn(x)g Fr(is)f Fn(P)7 b Fr(.)16 b(Then)g([)p Fn(K)i Fr(:)c Fn(k)r Fr(\()p Fn(x)p Fr(\)])g(=)f(deg)18 b(\(p)q(ole)e(divisor)g(of)g Fn(x)p Fr(\))i(=)e(1.)0 911 y Fp(Lecture)k(20.)c Fr(\(Octob)q(er)h(13,)f(1995\).)0 981 y(Scrib)q(e:)g(Dylan)g(Th)o(urston)0 1098 y(Plan)g(for)g(the)h (remainder)d(of)j(the)g(course:)50 1168 y Fm(\017)25 b Fr(\014elds)15 b(of)i(lo)o(w)f(gen)o(us)f(\(to)i(see)g(Riemann-Ro)q (c)o(h)d(at)j(w)o(ork\))50 1237 y Fm(\017)25 b Fr(line)19 b(bundles)f(/)h(linear)f(systems)h(/)h(maps)e(to)i(pro)s(jectiv)o(e)f (space)g(\(in)g(particular,)f(em)o(b)q(eddings)100 1307 y(of)g(a)g(curv)o(e)f(in)g Fp(R)437 1289 y Fk(n)465 1307 y Fr(\).)h(\(In)g(some)f(sense,)g(the)h(coming)f(computations)g(are)g (sp)q(ecial)h(cases)f(of)i(line)100 1377 y(bundles.\))50 1447 y Fm(\017)25 b Fr(connection)16 b(b)q(et)o(w)o(een)g(K\177)-25 b(ahler)15 b(and)h(W)l(eil)h(di\013eren)o(tials)e(\(with)h(pro)q(ofs)g (omitted)h(at)g(times\))50 1516 y Fm(\017)25 b Fr(gen)o(us)15 b(form)o(ula)g(of)h(Hurewicz)h(/)f(extensions)g(of)h(lo)q(cal)g (\014elds)50 1586 y Fm(\017)25 b Fn(\020)t Fr(-functions)14 b(of)j(curv)o(es)f(o)o(v)o(er)f(\014nite)i(\014elds)50 1656 y Fm(\017)25 b Fr(Riemann)15 b(h)o(yp)q(othesis)g(for)h(curv)o(es) g(o)o(v)o(er)g(\014nite)g(\014elds)50 1726 y Fm(\017)25 b Fr(and)15 b(\014nally)l(,)h(the)h(relation)f(of)g(the)h(last)f(t)o(w) o(o)h(items)f(to)h(co)q(ding)f(theory)l(.)0 1795 y Fp(Fields)21 b(of)d(gen)n(us)h Fr(0)p Fp(.)100 1865 y Fr(F)l(or)c(this)h(lecture,)g (assume)g Fn(k)f Fm(\032)e Fn(K)21 b Fr(and)890 1852 y(\026)889 1865 y Fn(k)12 b Fm(\\)g Fn(K)17 b Fr(=)d Fn(k)r Fr(.)100 1935 y(If)h Fn(K)j Fr(=)13 b Fn(k)r Fr(\()p Fn(t)p Fr(\),)j(then)f Fn(K)k Fr(has)14 b(gen)o(us)g(0.)h(Con)o(v)o (ersely)l(,)f(if)h Fn(K)k Fr(has)14 b(gen)o(us)g(0)h(and)g(there)g (exists)g Fn(P)21 b Fm(2)14 b Fn(C)0 2005 y Fr(with)j(deg)8 b Fn(P)21 b Fr(=)14 b(1)i(\(or,)h(equiv)m(alen)o(tly)l(,)f Fn(C)t Fr(\()p Fn(k)r Fr(\))d Fm(6)p Fr(=)h Fm(;)p Fr(\))j(then)g Fn(K)1131 1991 y Fm(\030)1131 2007 y Fr(=)1184 2005 y Fn(k)r Fr(\()p Fn(t)p Fr(\).)g(So)f(the)h(in)o(teresting)e(case)i(will) f(b)q(e)0 2074 y(when)g Fn(C)t Fr(\()p Fn(k)r Fr(\))d(=)h Fm(;)p Fr(.)0 2144 y Fp(F)-5 b(unction)20 b(\014elds)g(of)e(conics.)100 2214 y Fr(No)o(w)e(pic)o(k)g(a)h(non-zero)e(\(and)h(usually)g (irreducible)e(o)o(v)o(er)i Fn(k)r Fr(\))g(quadratic)g(p)q(olynomial) 469 2323 y Fn(f)j Fr(=)565 2275 y Fh(X)645 2323 y Fr(0)14 b Fm(\024)f Fn(i)h Fm(\024)g Fn(j)i Fm(\024)e Fr(2)p Fn(a)961 2330 y Fk(ij)996 2323 y Fn(X)1037 2330 y Fk(i)1054 2323 y Fn(X)1095 2330 y Fk(j)1130 2323 y Fm(2)g Fn(k)r Fr([)p Fn(X)1260 2330 y Fl(0)1282 2323 y Fn(;)8 b(X)1345 2330 y Fl(1)1368 2323 y Fn(;)g(X)1431 2330 y Fl(2)1454 2323 y Fr(])p Fn(:)0 2429 y Fr(If)16 b Fn(f)22 b Fr(is)15 b(reducible,)f(the)i(zero-set)f(of)h(eac)o(h)f(factor)h(is)f(a)h(line;) f(the)h(t)o(w)o(o)g(lines)e(are)i(either)f(degenerate)h(or)0 2498 y(in)o(tersect)d(in)h(a)g(p)q(oin)o(t.)f(If)h(they)h(in)o(tersect) e(in)h(a)g(p)q(oin)o(t,)f(then)h(the)g(v)m(ariet)o(y)g(is)g(singular.)e Fn(f)20 b Fr(is)13 b(irreducible)0 2568 y(i\013)j(the)h(zero-set)f(is)g (non-singular,)e(i.e.,)i Fn(f)5 b Fr(,)17 b Fn(f)856 2575 y Fl(0)879 2568 y Fr(,)f Fn(f)933 2575 y Fl(1)956 2568 y Fr(,)g Fn(f)1010 2575 y Fl(2)1050 2568 y Fr(ha)o(v)o(e)g(no)g (common)f(ro)q(ot;)i(or)f(c)o(har)7 b Fn(k)15 b Fr(=)f(2)i(and)730 2670 y Fn(f)k Fr(=)13 b Fn(aX)897 2649 y Fl(2)893 2682 y(0)931 2670 y Fr(+)e Fn(bX)1047 2649 y Fl(2)1043 2682 y(1)1081 2670 y Fr(+)g Fn(cX)1198 2649 y Fl(2)1194 2682 y(2)950 2790 y Fr(41)p eop %%Page: 42 42
42 41 bop 0 50 a Fr(with)16 b Fn(a;)8 b(b;)g(c)18 b Fr(satisfying)680 120 y(\()p Fn(a)c Fr(:)g Fn(b)g Fr(:)g Fn(c)p Fr(\))g Fm(2)g Fp(P)971 99 y Fl(2)993 120 y Fr(\()p Fn(k)r Fr(\))d Fm(\000)g Fp(P)1159 99 y Fl(2)1182 120 y Fr(\()p Fn(k)1229 99 y Fl(2)1251 120 y Fr(\))0 223 y(\(whic)o(h)17 b(is)g(to)h(sa)o(y)l (,)f(the)h(ratios)f(b)q(et)o(w)o(een)g Fn(a)p Fr(,)h Fn(b)p Fr(,)g(and)f Fn(c)g Fr(are)g(not)h(all)f(squares\).)g(The)g (function)g(\014eld)g Fn(K)0 294 y Fr(of)g Fn(f)22 b Fr(is)16 b(the)h(\014eld)f(generated)g(b)o(y)g Fn(x)e Fr(=)744 274 y Fk(X)777 279 y Fd(1)p 744 283 53 2 v 744 312 a Fk(X)777 317 y Fd(0)819 294 y Fr(and)i Fn(y)g Fr(=)1014 274 y Fk(X)1047 279 y Fd(2)p 1014 283 V 1014 312 a Fk(X)1047 317 y Fd(0)1089 294 y Fr(in)g Fn(Q)p Fr(\()p Fn(k)r Fr([)p Fn(x)1275 301 y Fl(0)1298 294 y Fn(;)8 b(X)1361 301 y Fl(1)1384 294 y Fn(;)g(X)1447 301 y Fl(2)1469 294 y Fr(])p Fn(=)p Fr(\()p Fn(f)d Fr(\)\).)0 369 y Fp(Remark.)15 b Fr(As)i(a)g(function)f(on)g Fn(k)d Fm(\002)d Fn(k)j Fm(\002)e Fn(k)r Fr(,)16 b Fn(f)22 b Fr(is)17 b(not)f(iden)o(tically)g (0.)g(\(Explicit)h(argumen)o(t:)e(Supp)q(ose)0 438 y Fn(f)22 b Fr(w)o(ere)16 b(0)h(as)f(a)h(function.)f(Then)g(plugging)f (in)h(1)p Fn(;)8 b Fr(0)p Fn(;)g Fr(0)14 b Fm(\))g Fn(a)1128 445 y Fl(00)1184 438 y Fr(=)g(0,)i(similarly)f(for)h Fn(a)1598 445 y Fl(11)1657 438 y Fr(and)g Fn(a)1780 445 y Fl(22)1823 438 y Fr(;)g(then)0 508 y(plugging)f(in)g(1)p Fn(;)8 b Fr(1)p Fn(;)g Fr(0)14 b Fm(\))g Fn(a)482 515 y Fl(01)538 508 y Fr(=)g(0,)h(similarly)g(for)g Fn(a)950 515 y Fl(12)1009 508 y Fr(and)g Fn(a)1131 515 y Fl(02)1174 508 y Fr(.\))h(So)g(b)o(y)g(a)g(linear)f(c)o(hange)g(of)h(v)m(ariables) 0 582 y(w)o(e)i(ma)o(y)f(assum)f Fn(f)5 b Fr(\(0)p Fn(;)j Fr(0)p Fn(;)g Fr(1\))17 b Fm(6)p Fr(=)f(0,)h(i.e.,)h Fn(a)771 589 y Fl(22)829 582 y Fm(6)p Fr(=)e(0.)h(Then)h(replace)f Fn(f)23 b Fr(b)o(y)1383 559 y Fk(f)p 1366 570 58 2 v 1366 599 a(a)1388 604 y Fd(22)1448 582 y Fr(to)18 b(mak)o(e)f Fn(a)1667 589 y Fl(22)1725 582 y Fr(=)f(1.)i(Then)0 656 y Fn(K)i Fr(is)d(an)f(extension)g(of)h Fn(k)r Fr(\()p Fn(x)p Fr(\))g(of)g(the)g(form)e Fn(k)r Fr(\()p Fn(x)p Fr(\)[)p Fn(y)r Fr(])j(where)e Fn(y)i Fr(satis\014es)320 783 y Fn(y)346 763 y Fl(2)380 783 y Fr(+)11 b(\(p)q(oly)l(.)17 b(in)f Fn(x)h Fr(of)f(deg)f Fm(\024)e Fr(1\))f Fm(\001)f Fn(y)i Fr(+)e(\(p)q(oly)l(.)16 b(in)h Fn(x)g Fr(of)f(deg)f Fm(\024)e Fr(2\))h(=)g(0)p Fn(:)256 b Fr(\()p Fm(\003)p Fr(\))0 910 y(Since)16 b(this)g(is)g(a)h(monic)e(and)h(irreducible)f(p) q(olynomial,)g Fn(K)21 b Fr(is)16 b(a)g(degree)g(2)h(extension)f(of)h Fn(k)r Fr(\()p Fn(x)p Fr(\).)100 980 y(No)o(w)f(w)o(e'll)g(sho)o(w)f (the)i(gen)o(us)e(of)i Fn(K)k Fr(is)16 b(0.)g(By)h(one)g(of)f(the)h (exercises,)467 1114 y Fn(K)g Fm(\023)d Fr(in)o(t.)i(closure)f(of)i Fn(k)r Fr([)p Fn(x)p Fr(])c(=)1052 1067 y Fh([)1042 1174 y Fk(n)p Fj(\025)p Fl(0)1126 1114 y Fn(L)p Fr(\()p Fn(nD)q Fr(\))i Fm(\023)f Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])p Fn(:)0 1295 y Fr(\(Here)k Fn(D)h Fr(is)e(the)h(divisor)e(so)i(that)g Fn(x)i Fr(=)f Fn(N)6 b Fm(\000)q Fn(D)14 b Fr(with)d Fn(N)17 b Fr(and)11 b Fn(D)j Fr(e\013ectiv)o(e.\))e(Also,)f Fn(k)r Fr([)p Fn(x;)d(y)r Fr(])14 b(=)g Fn(k)r Fr([)p Fn(x)p Fr(])q(+)q Fn(y)r(k)r Fr([)p Fn(x)p Fr(])0 1364 y(is)i(a)h(free)f(mo)q(dule)g(of)h(rank)f(2)g(o)o(v)o(er)g Fn(k)r Fr([)p Fn(x)p Fr(].)100 1434 y(The)j(basic)h(tec)o(hnique)f (will)h(b)q(e)g(to)g(use)f(Riemann-Ro)q(c)o(h,)f(whic)o(h)g(in)i(this)f (case)h(sa)o(ys)f(that,)h(for)0 1504 y Fn(n)14 b(>)-8 b(>)13 b Fr(0,)632 1573 y(dim)715 1580 y Fk(k)748 1573
y Fn(L)p Fr(\()p Fn(nD)q Fr(\))i(=)e(deg)q(\()p Fn(nD)q Fr(\))959 1607 y Fh(|)p 981 1607 49 6 v 49 w({z)p 1074 1607 V 49 w(})1030 1653 y Fl(2)p Fk(n)1154 1573 y Fr(+1)e Fm(\000)g Fn(g)r(:)542 b Fr(\()p Fm(\003\003)p Fr(\))0 1743 y Fp(Claim.)17 b Fr(If)g(ord)303 1750 y Fk(P)345 1743 y Fn(x)d(<)g Fr(0)i(\()p Fn(x)i Fr(has)d(a)i(p)q(ole)f(at)h Fn(P)7 b Fr(,)17 b(so)f Fn(D)f Fm(\025)f Fn(P)7 b Fr(\))17 b(then)f(ord)1343 1750 y Fk(P)1385 1743 y Fn(y)g Fm(\025)d Fr(ord)1550 1750 y Fk(P)1591 1743 y Fn(x)p Fr(.)0 1813 y Fp(Pro)r(of.)h Fr(Lo)q(ok)j(at)f(the)h(relation)e(\()p Fm(\003)p Fr(\).)h(If)h Fn(y)h Fr(has)d(a)h(large)g(p)q(ole,)g(then)g (the)g(\014rst)f(term)h(dominates)f(when)0 1883 y(y)o(ou)h(tak)o(e)h (the)g(order)e(at)i Fn(p)p Fr(,)g(a)f(con)o(tradiction.)100 1953 y(F)l(rom)f(the)h(claim,)g(w)o(e)g(immediately)g(ha)o(v)o(e)g Fn(y)g Fm(2)e Fn(L)p Fr(\()p Fn(D)q Fr(\))k(and)e(therefor)573 2080 y Fn(L)p Fr(\()p Fn(nD)q Fr(\))f Fm(\023)f Fn(k)r Fr([)p Fn(x)p Fr(])869 2087 y Fl(deg)p Fj(\024)p Fk(n)998 2080 y Fr(+)d Fn(y)i Fm(\001)e Fn(k)r Fr([)p Fn(x)p Fr(])1194 2087 y Fl(deg)p Fj(\024)p Fk(n)p Fj(\000)p Fl(1)1363 2080 y Fn(:)0 2207 y Fr(The)16 b(dimension)e(o)o(v)o(er)h Fn(k)i Fr(of)f(the)g(righ)o(t)f(hand)g(side)g(is)g(\()p Fn(n)10 b Fr(+)g(1\))g(+)f Fn(n)14 b Fr(=)g(2)p Fn(n)9 b Fr(+)g(1.)16 b(Th)o(us,)e(b)o(y)i(\()p Fm(\003\003)p Fr(\),)g Fn(g)g Fr(=)d(0,)573 2334 y Fn(L)p Fr(\()p Fn(nD)q Fr(\))i(=)f Fn(k)r Fr([)p Fn(x)p Fr(])869 2341 y Fl(deg)p Fj(\024)p Fk(n)998 2334 y Fr(+)d Fn(y)i Fm(\001)e Fn(k)r Fr([)p Fn(x)p Fr(])1194 2341 y Fl(deg)p Fj(\024)p Fk(n)p Fj(\000)p Fl(1)1363 2334 y Fn(;)0 2461 y Fr(and)17 b(the)g(in)o(tegral) f(closure)h(of)g Fn(k)r Fr([)p Fn(x)p Fr(])g(is)743 2423 y Fh(S)793 2461 y Fn(L)p Fr(\()p Fn(nD)q Fr(\))f(=)f Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(].)17 b(\(Notice)i(that)e(only)g(the)h (trivial)f(part)g(of)0 2531 y(the)g(exercise)f(w)o(as)g(needed.\))100 2600 y(W)l(e)g(ha)o(v)o(e)g(pro)o(v)o(ed)f(that)i Fp(function)j (\014elds)f(of)f(conics)i(ha)n(v)n(e)g(gen)n(us)f(0)p Fr(.)0 2670 y Fq(Question:)e Fr(Do)g(w)o(e)f(get)h(an)o(y)f(function)g (\014elds)g(b)q(esides)g(the)h(one)f(w)o(e)g(found)g(earlier?)950 2790 y(42)p eop %%Page: 43 43 43 42 bop 100 50 a Fr(Remem)o(b)q(er)14 b(that)i(w)o(e)f(sho)o(w)o(ed)f (that)i(if)g Fn(k)i Fr(has)d(gen)o(us)f Fn(g)h Fr(=)f(0,)i(then)f Fn(C)t Fr(\()p Fn(k)r Fr(\))e Fm(6)p Fr(=)h Fm(;)g(,)f(9)p Fn(t)h Fr(:)g Fn(K)j Fr(=)d Fn(k)r Fr(\()p Fn(t)p Fr(\).)0 120 y(Y)l(ou)i(should)f(think)612 189 y Fn(C)t Fr(\()p Fn(k)r Fr(\)\\=")o Fm(f)p Fn(P)21 b Fm(2)14 b Fp(P)970 169 y Fl(2)992 189 y Fr(\()p Fn(k)r Fr(\))h(:)e Fn(f)5 b Fr(\()p Fn(P)i Fr(\))15 b(=)f(0)p Fm(g)p Fn(:)0 278 y Fr(In)h(particular,)f(the)i(t)o(w)o(o)f(sets)h(ha)o(v)o(e)f(the)h (same)f(size.)g(\()p Fq(Exer)m(cise:)i Fr(Chec)o(k)f(this.\))f(Th)o(us) f Fn(K)k Fm(')c Fn(k)r Fr(\()p Fn(t)p Fr(\))i Fm(,)f Fn(f)0 348 y Fr(has)h(a)g(non-trivial)f(zero)i(o)o(v)o(er)e Fn(k)r Fr(.)0 429 y Fq(Example:)i Fr(The)f(curv)o(e)g(with)g Fn(k)f Fr(=)f Fp(R)p Fr(,)i Fn(f)k Fr(=)13 b Fn(X)872 411 y Fl(2)868 441 y(0)905 429 y Fr(+)e Fn(X)1000 411 y Fl(2)996 441 y(1)1033 429 y Fr(+)f Fn(X)1127 411 y Fl(2)1123 441 y(2)1166 429 y Fr(has)15 b(function)h(\014eld)g Fn(K)i Fr(=)13 b Fp(R)p Fr(\()p Fn(x;)8 b(y)r Fr(\))p Fn(;)g(x)1876 411 y Fl(2)1911 429 y Fr(+)0 499 y Fn(y)26 481 y Fl(2)62 499 y Fr(=)14 b Fm(\000)p Fr(1)i(whic)o(h)g(has)g(gen)o (us)f(0)h(but)h(not)f(isomorphic)e(to)j Fp(R)p Fr(\()p
Fn(x)p Fr(\).)100 568 y(There)g(are)g(man)o(y)g(more)g(examples)g(of)h (non-isomorphi)o(c)d(conics)j(o)o(v)o(er)f Fp(Q)p Fr(:)g Fn(f)24 b Fr(ma)o(y)17 b(fail)h(to)g(ha)o(v)o(e)0 638 y(a)f(ro)q(ot)f Fn(p)p Fr(-adically)g(for)g(di\013eren)o(t)g Fn(p)p Fr(.)100 719 y(In)h(particular,)e(if)i Fn(k)g Fr(=)551 706 y(\026)549 719 y Fn(k)r Fr(,)g(then)g Fn(C)t Fr(\()p Fn(k)r Fr(\))e Fm(6)p Fr(=)f Fm(;)j Fr(\(since)g(for)g(all)g Fn(P)22 b Fm(2)15 b Fn(C)t Fr(,)i(deg)8 b Fn(P)22 b Fr(=)15 b(1\))i(and)g(all)g(conics)0 789 y(are)k(rational.)f(There)h(is)g(a)g (direct)g(geometric)g(pro)q(of)g(of)g(this:)g(giv)o(en)g(an)o(y)g(p)q (oin)o(t)g Fn(P)28 b Fr(on)21 b(the)g(conic,)0 858 y(parametrize)14 b(all)h(other)h(p)q(oin)o(ts)e Fn(Q)i Fr(b)o(y)f(the)h(slop)q(e)f(of)h (the)g(\(unique\))f(line)g(passing)f(through)g Fn(P)23 b Fr(and)15 b Fn(Q)p Fr(.)0 926 y(If)j(w)o(e)g(c)o(ho)q(ose)f(co)q (ordinates)g(so)h(that)g Fn(P)25 b Fr(is)17 b(at)h(\(1)f(:)e(0)h(:)g (0\))i(and)g(set)g Fn(x)e Fr(=)1399 906 y Fk(X)1432 911 y Fd(1)p 1399 915 53 2 v 1399 943 a Fk(X)1432 948 y Fd(0)1475 926 y Fr(and)h Fn(y)h Fr(=)1676 906 y Fk(X)1709 911 y Fd(2)p 1676 915 V 1676 943 a Fk(X)1709 948 y Fd(0)1734 926 y Fr(,)g(then)g(the)0 997 y(slop)q(e)e Fn(t)e Fr(=)217 975 y Fk(y)p 217 985 23 2 v 217 1014 a(x)262 997 y Fr(generates)i(the)h (function)f(\014eld)g Fn(K)t Fr(.)100 1066 y(It)h(is)f(a)g(non-trivial) f(fact)i(that)g(if)g Fn(k)h Fr(is)e Fq(\014nite)h Fr(all)f(conics)g (are)g(rational.)0 1136 y Fp(Theorem.)j Fr(If)i Fn(K)k Fm(\033)20 b Fn(k)j Fr(is)d(a)h(function)f(\014eld)g(of)h(gen)o(us)e (0,)i(then)g(it)g(is)f(isomorphic)e(to)k Fn(k)r Fr(\()p Fn(t)p Fr(\))f(or)f(the)0 1206 y(function)c(\014eld)g(of)h(a)f(conic.)0 1276 y Fp(Pro)r(of.)e Fr(Let)j(at)g(deg)d(:)g(Div)517 1283 y Fk(K)569 1276 y Fm(!)f Fp(Z)p Fr(.)k(The)f(image)f(is)h(a)g (non-zero)f(subgroup)f(of)i Fp(Z)g Fr(con)o(taining)f(2)p Fn(g)d Fm(\000)e Fr(2,)0 1345 y(the)19 b(degree)f(of)h(the)f(canonical) g(divisor.)f(So)h(deg)q(\(Div)1035 1352 y Fk(K)1073 1345 y Fr(\))g(=)e Fn(d)p Fp(Z)j Fr(for)f(some)g Fn(d)f Fm(2)g(f)p Fr(1)p Fn(;)8 b Fr(2)p Fm(g)p Fr(.)18 b(\(In)h(general,)0 1415 y Fn(d)8 b Fm(j)g Fr(2)p Fn(g)13 b Fm(\000)e Fr(2.)16 b(This)g(is)g(useful)g(except)h(when)f Fn(g)f Fr(=)f(1.\))100 1485 y(Cho)q(ose)f(an)o(y)g Fn(D)j Fr(with)d(deg)c Fn(D)15 b Fr(=)f Fn(d)p Fr(.)g(Then,)f(b)o(y)g(Riemann-Ro)q(c)o(h,)e Fn(L)p Fr(\()p Fn(D)q Fr(\))k(has)e(dimension)f(deg)c Fn(D)f Fr(+)0 1555 y(1)p Fn(g)r(e)p Fr(2)14 b Fn(>)g Fr(1.)i(So)h(there)f(is)h(a)f Fn(f)k Fm(6)p Fr(=)14 b(0)g Fm(2)h Fn(L)p Fr(\()p Fn(D)q Fr(\))j(and)e(\()p Fn(f)5 b Fr(\))12 b(+)f Fn(D)16 b Fm(\025)e Fr(0.)j(So)f(assume)f(that)i Fn(D)i Fr(is)d(e\013ectiv)o(e)i(and)0 1624 y(therefore)e Fn(L)p Fr(\()p Fn(D)q Fr(\))f Fm(\033)f Fn(k)h Fr(=)f Fn(L)p Fr(\(0\).)0 1694 y Fp(Case)19 b Fn(d)14 b Fr(=)g(1)p Fp(:)i Fn(D)i Fr(is)e Fq(prime)p Fr(,)h(of)g(degree)f(1,)g(so)g Fn(K)i Fr(=)c Fn(k)r Fr(\()p Fn(x)p Fr(\))j(\(for)g(an)o(y)f Fn(x)e Fm(2)g Fn(L)p Fr(\()p Fn(D)q Fr(\))e Fm(\000)f Fn(k)r Fr(\).)0 1764 y Fp(Case)22 b Fn(d)c Fr(=)f(2)p Fp(:)i Fr(W)l(e)g(ma)o(y)f(assume)g Fn(D)j Fr(is)d(prime.)g (\(Otherwise)g Fn(D)h Fr(=)f Fn(P)h Fr(+)13 b Fn(Q)19 b Fr(with)g Fn(P)26 b Fr(and)18 b Fn(Q)h Fr(prime,)0 1834 y(con)o(tradicting)c(the)i(assumption)d(that)j Fn(d)d Fr(=)g(1.\))i(Then)g Fn(L)p Fr(\()p Fn(D)q Fr(\))i(is)e(3-dimensional:) 712 1928 y Fn(L)p Fr(\()p Fn(D)q Fr(\))f(=)e Fn(k)g Fr(+)e Fn(k)h Fm(\001)f Fn(x)h Fr(+)f Fn(k)h Fm(\001)f Fn(y)r(:)0 2023 y(x)20 b Fr(has)f Fn(D)j Fr(as)d(a)h(denominator)e(divisor)g(of)i (degree)g(2,)f(so)g([)p Fn(K)k Fr(:)c Fn(k)r Fr(\()p
Fn(x)p Fr(\)])h(=)f(2.)h(Since)f Fn(y)j Fr(also)d(has)g Fn(D)i Fr(as)0 2093 y(a)e(denominator)d(divisor)i(of)h(degree)f(2,)g Fn(y)25 b(=)-30 b Fm(2)17 b Fn(k)r Fr(\()p Fn(x)p Fr(\))j(\(as)e(all)h (p)q(o)o(w)o(ers)e(of)i Fn(x)g Fr(ha)o(v)o(e)f(a)g(larger)g(m)o (ultiple)f(of)0 2162 y Fn(D)k Fr(as)e(denominator)f(divisor\))g(and)h Fn(K)j Fr(=)c Fn(k)r Fr(\()p Fn(x;)8 b(y)r Fr(\).)21 b(W)l(e)e(m)o(ust)g(sho)o(w)f(that)i Fn(y)h Fr(satis\014es)e(a)g (quadratic.)0 2232 y(Consider)c(the)i Fn(k)r Fr(-v)o(ector)e(spaces)725 2302 y Fn(L)p Fr(\(0\))g Fm(\032)e Fn(L)p Fr(\()p Fn(D)q Fr(\))i Fm(\032)f Fn(L)p Fr(\(2)p Fn(D)q Fr(\))p Fn(:)0 2391 y Fr(These)19 b(ha)o(v)o(e)g(dimensions,)e(resp)q(ectiv)o(ely)l(,) i(1,)h(3,)f(and)g(5,)g(with)h(new)f(elemen)o(ts)g(of)h(a)f(basis)g(giv) o(en)g(b)o(y)l(,)0 2461 y(resp)q(ectiv)o(ely)l(,)d(1;)f Fn(x)i Fr(and)e Fn(y)r Fr(;)h(and)f Fn(x)656 2443 y Fl(2)695 2461 y Fr(and)g Fn(xy)r Fr(.)i(Since)e Fn(y)1031 2443 y Fl(2)1067 2461 y Fm(2)f Fn(L)p Fr(\(2)p Fn(D)q Fr(\),)j(it)g(can)e(b) q(e)h(expressed)f(in)h(terms)f(of)0 2531 y(this)h(basis)f(and)h (satis\014es)g(a)g(quadratic)g(equation)h(o)o(v)o(er)e Fn(k)r Fr(\()p Fn(x)p Fr(\).)100 2600 y([This)g(pro)q(of)h(will)h(b)q (e)f(seen)h(in)f(a)g(b)q(etter)h(p)q(ersp)q(ectiv)o(e)g(with)f(line)g (bundles.])0 2670 y Fq(R)m(efer)m(enc)m(e:)i Fr(Miranda,)d Fq(A)o(lgebr)m(aic)j(Curves)g(and)g(R)o(iemann)h(Surfac)m(es)p Fr(.)950 2790 y(43)p eop %%Page: 44 44 44 43 bop 0 50 a Fp(Lecture)20 b(21)c Fr(\(Octob)q(er)h(16,)f(1995\).)0 120 y(Scrib)q(e:)g(Matt)h(Bak)o(er)100 244 y(Recall)h(that)i(in)e(the)i (last)f(lecture,)g(w)o(e)g(sho)o(w)o(ed)e(that)i(if)h Fn(K)i Fm(\033)17 b Fn(k)k Fr(is)e(a)g(function)g(\014eld)f(of)h(gen)o (us)0 314 y(zero,)d(then)h(it)f(is)h(isomorphic)d(to)j(either)f Fn(k)r Fr(\()p Fn(t)p Fr(\))h(or)f(the)h(function)f(\014eld)g(of)g(a)h (conic.)100 424 y(Here)f(is)g(the)h(last)g(detail)f(in)g(that)h (argumen)o(t:)100 513 y(W)l(e)i(had)g Fn(L)p Fr(\(0\))g Fm(\032)g Fn(L)p Fr(\()p Fn(D)q Fr(\))g Fm(\032)g Fn(L)p Fr(\(2)p Fn(D)q Fr(\),)h(and)f(w)o(e)g(claimed)g(that)h Fn(x)1327 495 y Fl(2)1369 513 y Fr(and)f Fn(xy)j Fr(w)o(ere)d(linearly) f(inde-)0 583 y(p)q(enden)o(t)f(elemen)o(ts)f(of)h Fn(L)p Fr(\(2)p Fn(D)q Fr(\),)h(where)f(1)p Fn(;)8 b(x;)g(y)20 b Fr(span)c Fn(L)p Fr(\()p Fn(D)q Fr(\).)i(T)l(o)f(see)g(this:)g(if)g (w)o(e)g(had)f(a)i(dep)q(endence)0 653 y(relation,)e(i.e.)g(if)h Fn(ax)377 635 y Fl(2)411 653 y Fr(+)11 b Fn(bxy)16 b Fm(2)f Fn(L)p Fr(\()p Fn(D)q Fr(\))p Fn(;)8 b(a;)g(b)16 b Fm(2)e Fn(k)r Fr(,)i(then)g Fn(ax)c Fr(+)f Fn(by)16 b Fm(2)e Fn(L)p Fr(\(0\),)j(so)f(a=b=0.)100 762 y(There)g(is)g(a)g (similar)f(story)h(for)h(\014elds)e(of)i(gen)o(us)e(one.)100 852 y(In)h(general,)f(what)i(w)o(e)f(do)h(is)f(lo)q(ok)h(at)f(the)h (map)792 971 y(deg)d(:)g Fn(D)q(iv)992 978 y Fk(K)1045 971 y Fm(!)g Fp(Z)p Fn(;)0 1089 y Fr(whic)o(h)i(con)o(tains)f(2)p Fn(g)d Fm(\000)f Fr(2)17 b(in)f(its)g(image)g(b)q(ecause)h(of)f(the)h (existence)g(of)g(the)g(canonical)e(divisor.)100 1179 y(In)20 b(the)h(case)f Fn(g)i Fr(=)e(0,)h(w)o(e)f(kno)o(w)g(the)h (degree)f(map)g(is)g(either)g(on)o(to)g(or)g(has)g(cok)o(ernel)g(of)g (order)0 1249 y(2.)h(But)i(a)e(priori,)f(for)h Fn(g)j(>)f Fr(0,)e(w)o(e)h(don't)f(kno)o(w)g(m)o(uc)o(h)f(else)h(ab)q(out)h(the)g (image)f(of)h(this)f(map.)g(An)0 1318 y(in)o(teresting)d(question)h(to) h(think)g(ab)q(out)f(\(but)h(ab)q(out)f(whic)o(h)g(w)o(e)g(w)o(on't)g (sa)o(y)g(a)h(whole)f(lot)g(more\))h(is)0 1388 y(the)d(folo)o(wing.)100 1498 y Fp(Problem)p Fr(:)f(Giv)o(en)i Fn(n)d Fm(2)h Fp(Z)619 1505 y Fk(>)p Fl(0)673 1498 y Fr(,)h(construct)g(a)h(\014eld)f Fn(k)i Fr(plus)e(a)h(function)f(\014eld)g(of)h(gen)o(us)e(1)i(o)o(v)o
(er)e Fn(k)0 1567 y Fr(suc)o(h)f(that)i(image\(deg\))g(=)f Fn(n)p Fp(Z)p Fr(.)100 1677 y(No)o(w,)h(let)i Fn(K)q(=k)g Fr(ha)o(v)o(e)e(gen)o(us)g(1,)h(and)f(supp)q(ose)g(that)h(the)g(degree) g(map)f(is)g Fp(surjectiv)n(e)p Fr(.)j(Recall:)0 1747 y(Riemann-Ro)q(c)o(h)14 b(sa)o(ys)i(that)h(\(if)g(w)o(e)f(let)h Fn(D)h Fr(b)q(e)f(a)g(divisor)e(of)i(degree)f Fn(m)p Fr(\))773 1865 y Fn(l)q Fr(\()p Fn(D)q Fr(\))f(=)f Fn(m;)8 b Fr(\()p Fn(m)14 b(>)f Fr(0\))783 1984 y Fn(l)q Fr(\()p Fn(D)q Fr(\))i(=)e(0)p Fn(;)8 b Fr(\()p Fn(m)14 b(<)g Fr(0\))0 2084 y(In)i(general,)g Fn(l)q Fr(\()p Fn(D)q Fr(\))f Fm(\024)e Fr(1+)k(deg\()p Fn(D)q Fr(\),)g(and)f(if)h(deg\()p Fn(D)q Fr(\)=0,)762 2203 y Fn(l)q Fr(\()p Fn(D)q Fr(\))e(=)e(1)p Fn(;)8 b Fr(\()p Fn(D)16 b Fr(=)e(\()p Fn(f)5 b Fr(\)\))755 2321 y Fn(l)q Fr(\()p Fn(D)q Fr(\))15 b(=)e(0)p Fn(;)8 b Fr(\()p Fn(D)16 b Fm(6)p Fr(=)e(\()p Fn(f)5 b Fr(\)\))p Fn(:)0 2421 y Fr(So)12 b(the)h(canonical)f(divisor,)g(whic)o(h)f(has)h (degree)h(2)p Fn(g)5 b Fm(\000)s Fr(2)13 b(=)h(0,)f(m)o(ust)e(b)q(e)i (equiv)m(alen)o(t)g(to)g(0)g(as)f(an)h(elemen)o(t)0 2491 y(of)k(the)g(divisor)e(class)h(group.)100 2600 y(No)o(w)j(c)o(ho)q(ose) f(a)i(divisor)e Fn(D)i Fr(with)f(deg\()p Fn(D)q Fr(\)=1.)h([Whic)o(h)f (w)o(e)g(can)f(do)h(b)o(y)g(the)h(assumption)d(that)0 2670 y(deg)f(is)g(surjectiv)o(e].)950 2790 y(44)p eop %%Page: 45 45 45 44 bop 100 50 a Fr(Then)13 b Fn(L)p Fr(\()p Fn(D)q Fr(\))i(has)e(dimension)f Fn(l)q Fr(\()p Fn(D)q Fr(\))j(=)f(1,)g(so)f (\()p Fn(f)5 b Fr(\))h(+)g Fn(D)16 b Fm(\025)e Fr(0)g(for)f(some)g (nonzero)g(elemen)o(t)h Fn(f)19 b Fm(2)14 b Fn(K)1913 32 y Fj(\003)1936 50 y Fr(.)0 120 y(Hence)20 b(there)f(is)g(an)h (e\013ectiv)o(e)g(divisor)e(of)i(degree)f(1,)g(i.e.,)g(a)h(prime)e (divisor)g Fn(P)1526 127 y Fl(0)1568 120 y Fr(with)i(deg)f Fn(P)1811 127 y Fl(0)1853 120 y Fr(=)g(1.)0 189 y(This)d(means)f(that)i Fn(P)406 196 y Fl(0)442 189 y Fm(2)d Fn(C)t Fr(\()p Fn(k)r Fr(\).)100 279 y(W)l(e)j(will)f(no)o(w)g(k)o(eep)h(this)f(p)q(oin)o(t)h Fn(P)758 286 y Fl(0)797 279 y Fr(\014xed)g(for)f(a)h(while)f([some)g(p) q(eople)h(who)f(w)o(ork)h(with)f(elliptic)0 349 y(curv)o(es)h(call)i (this)f(p)q(oin)o(t)g(0,)g(b)q(ecause)g(it)h(acts)f(as)g(the)h(iden)o (tit)o(y)f(elemen)o(t)g(in)g(the)g(group)f(la)o(w)h(on)g(suc)o(h)0 418 y(curv)o(es].)100 508 y(An)12 b Fp(ellipt)q(ic)19 b(curv)n(e)13 b Fr(can)g(b)q(e)f(de\014ned)g(as)g(a)h(function)f (\014eld)g(of)h(gen)o(us)f(one)g(plus)g(a)g(p)q(oin)o(t)h(of)f(degree)0 578 y(1)j(in)g Fn(C)t Fr(\()p Fn(k)r Fr(\).)g(As)h(w)o(as)e(the)i(case) f(for)g(\014elds)g(of)g(gen)o(us)g(zero,)g(w)o(e)g(can)g(extract)h (some)f(useful)g(information)0 648 y(b)o(y)h(writing)g(out)g(a)h(c)o (hain)e(of)i(v)o(ector)g(spaces)723 771 y Fn(L)757 778 y Fl(0)793 771 y Fm(\032)c Fn(L)879 778 y Fl(1)915 771 y Fm(\032)h Fn(L)1002 778 y Fl(2)1033 771 y Fn(:)8 b(:)g(:)14 b Fm(\032)f Fn(L)1191 778 y Fl(6)1213 771 y Fn(;)0 895 y Fr(where)24 b Fn(L)186 902 y Fk(n)240 895 y Fr(=)j Fn(L)p Fr(\()p Fn(nP)421 902 y Fl(0)443 895 y Fr(\))e(has)f(dimension)e (1)j(for)f Fn(n)j Fr(=)g(0)d(and)g(dimension)e Fn(n)j Fr(for)f Fn(n)j Fr(=)f(1)p Fn(;)8 b Fr(2)p Fn(;)g(:)g(:)g(:)h(;)f Fr(6)0 964 y(\(Riemann-Ro)q(c)o(h\).)14 b(A)i(searc)o(h)f(for)g (explicit)h(elemen)o(ts)f(of)h(these)g(v)o(ector)g(spaces)f(yields)g (the)h(follo)o(wing)0 1034 y(mem)o(b)q(ers:)883 1104 y(1)e Fm(2)g Fn(L)p Fr(\(0\))854 1206 y Fn(x)h Fm(2)f Fn(L)p Fr(\(2)p Fn(P)1054 1213 y Fl(0)1076 1206 y Fr(\))856 1308 y Fn(y)i Fm(2)e Fn(L)p Fr(\(3)p Fn(P)1053 1315 y Fl(0)1075 1308 y Fr(\))843 1410 y Fn(x)871 1389 y Fl(2)908 1410 y Fm(2)g Fn(L)p Fr(\(4)p Fn(P)1065 1417 y Fl(0)1087
1410 y Fr(\))841 1512 y Fn(xy)j Fm(2)d Fn(L)p Fr(\(5)p Fn(P)1067 1519 y Fl(0)1089 1512 y Fr(\))801 1614 y Fn(y)827 1593 y Fl(2)849 1614 y Fn(;)8 b(x)899 1593 y Fl(3)936 1614 y Fm(2)14 b Fn(L)p Fr(\(6)p Fn(P)1093 1621 y Fl(0)1116 1614 y Fr(\))p Fn(:)0 1716 y Fr(Note)k(that)f(here)f Fn(x)h Fr(has)g(a)f(p)q(ole)h(of)g(order)f(exactly)i(2)e(at)h Fn(P)1114 1723 y Fl(0)1137 1716 y Fr(,)f(and)h(no)f(p)q(ole)h(an)o (ywhere)f(else.)g Fn(x)h Fr(is)g(also)0 1785 y(unique,)f(up)g(to)h(a)f (linear)g(transformation.)100 1875 y(By)h(6-dimensionalit)o(y)d(of)i Fn(L)p Fr(\(6)p Fn(P)718 1882 y Fl(0)741 1875 y Fr(\),)h(there)f(is)g (a)h(linear)e(relation)531 1999 y Fn(y)557 1978 y Fl(2)591 1999 y Fr(+)c Fn(a)667 2006 y Fl(1)689 1999 y Fn(xy)j Fr(+)d Fn(a)831 2006 y Fl(3)854 1999 y Fn(y)16 b Fr(=)d Fn(x)974 1978 y Fl(3)1008 1999 y Fr(+)e Fn(a)1084 2006 y Fl(2)1107 1999 y Fn(x)1135 1978 y Fl(2)1168 1999 y Fr(+)g Fn(a)1244 2006 y Fl(4)1267 1999 y Fn(x)g Fr(+)g Fn(a)1382 2006 y Fl(6)1405 1999 y Fn(;)0 2122 y Fr(where)22 b(w)o(e)f(ha)o(v)o(e)h(normalized)e(the)i(co)q(e\016cien)o(ts)g(of)g Fn(y)1046 2104 y Fl(2)1069 2122 y Fn(;)8 b(x)1119 2104 y Fl(3)1164 2122 y Fr(to)22 b(mak)o(e)g(them)g(b)q(oth)g(equal)g(to)g (1)g([see)0 2192 y(Knapp's)e('Elliptic)g(Curv)o(es')h(for)g(a)g (detailed)g(description)f(of)i(this)f(normalization,)e(as)i(w)o(ell)g (as)g(an)0 2262 y(explanation)16 b(of)h(the)f(terminology)g(for)g(the)h (co)q(e\016cien)o(ts].)100 2371 y(Notice)j(that)g([)p Fn(K)i Fr(:)c Fn(k)r Fr(\()p Fn(x)p Fr(\)])i(=)e(2,)h(and)g(as)g(in)g (one)g(of)h(the)f(homew)o(ork)f(problems,)f(one)j(sees)f(that)0 2441 y Fn(e)p Fr(\()p Fn(P)74 2448 y Fl(0)97 2441 y Fn(=)p Fm(1)p Fr(\))k(=)g(2,)e(since)h Fn(v)488 2448 y Fk(P)513 2453 y Fd(0)535 2441 y Fr(\()p Fn(x)p Fr(\))j(=)d Fm(\000)p Fr(2)g(and)f Fn(v)899 2448 y Fj(1)942 2441 y Fr(\()p Fn(x)p Fr(\))j(=)f Fm(\000)p Fr(1.)e(Also,)h(since)f Fn(v)1476 2448 y Fk(P)1501 2453 y Fd(0)1524 2441 y Fr(\()p Fn(y)r Fr(\))j(=)e Fm(\000)p Fr(3,)g(w)o(e)g(ha)o(v)o(e)0 2511 y Fn(y)16 b Fm(62)e Fn(k)r Fr(\()p Fn(x)p Fr(\),)j(so)f Fn(K)i Fr(=)c Fn(k)r Fr(\()p Fn(x;)8 b(y)r Fr(\).)100 2600 y(It)22 b(is)f(also)g(easy)h(to)h(use)e(a)h(rami\014cation)e (argumen)o(t)g(to)i(establish)f(the)h(irreducibilit)o(y)e(of)i(the)0 2670 y(linear)16 b(relation)f(b)q(et)o(w)o(een)i Fn(x)g Fr(and)f Fn(y)r Fr(.)950 2790 y(45)p eop %%Page: 46 46 46 45 bop 100 50 a Fr(Notice)23 b(that)f(if)h(w)o(e)f(write)g(do)o(wn)f Fn(a)802 57 y Fl(1)825 50 y Fn(;)8 b(a)873 57 y Fl(3)896 50 y Fn(;)g(a)944 57 y Fl(2)967 50 y Fn(;)g(a)1015 57 y Fl(4)1037 50 y Fn(;)g(a)1085 57 y Fl(6)1108 50 y Fr(,)22 b(then)h(w)o(e)f(can)g(determine)f(if)i(w)o(e)f(get)g(an)0 120 y(elliptic)16 b(curv)o(e)f(or)g(not)h(b)o(y)f(lo)q(oking)h(at)g (whether)g Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(])15 b(is)h(the)g(in)o (tegral)f(closure)f(of)i Fn(k)r Fr([)p Fn(x)p Fr(])g(in)f Fn(K)t Fr(.)h(This)0 189 y(is,)g(b)o(y)g(the)h(homew)o(ork,)e(some)h (sort)g(of)g(p)q(olynomial)g(condition)f(that)i(can)g(b)q(e)g(easily)f (v)o(eri\014ed.)100 280 y(T)l(o)e(elab)q(orate)g(on)h(this)f(sligh)o (tly)l(,)f(note)i(that)g(if)g Fn(A)g Fr(\()g(=)f(in)o(tegral)f(closure) h(of)h Fn(k)r Fr([)p Fn(x)p Fr(])f(in)g Fn(K)t Fr(\),)h(whic)o(h)f(is)0 350 y(equal)i(to)h Fm([)225 357 y Fk(n)p Fj(\025)p Fl(0)303 350 y Fn(L)p Fr(\()p Fn(nP)418 357 y Fl(0)441 350 y Fr(\),)f(is)g (strictly)g(bigger)g(than)f Fn(k)r Fr([)p Fn(x;)8 b(y)r Fr(],)17 b(then)f(there)g(exists)g Fn(z)g Fm(2)e Fn(L)p Fr(\()p Fn(nP)1716 357 y Fl(0)1739 350 y Fr(\))d Fm(\000)f Fn(k)r Fr([)p Fn(x;)e(y)r Fr(])0 420 y(for)22 b(some)g
Fn(n)p Fr(,)h(and)e(b)o(y)i(subtracting)e(appropriate)g(com)o (binations)f(of)j Fn(x;)8 b(y)r Fr(,)23 b(w)o(e)g(ma)o(y)f(assume)f (that)0 489 y Fn(z)16 b Fm(2)e Fn(L)p Fr(\()p Fn(P)171 496 y Fl(0)194 489 y Fr(\).)j(Then)f Fn(z)j Fr(generates)d(the)h(whole) f(function)g(\014eld,)g(so)g(the)g(gen)o(us)g(is)g(zero.)100 600 y(W)l(e)i(no)o(w)f(mo)o(v)o(e)g(on)h(to)g(a)g(di\013eren)o(t)f (topic:)g(line)h(bundles,)e(linear)h(systems,)g(and)g(maps)g(to)h(pro-) 0 670 y(jectiv)o(e)f(space.)100 760 y(Notice)g(the)g(follo)o(wing)e (pattern)h(in)g(our)g(classi\014cation)f(of)i(lo)o(w-gen)o(us)d (function)i(\014elds:)100 851 y Fn(g)f Fr(=)f(0)f(:)h Fn(L)p Fr(\(0\))g Fm(\032)g Fn(L)p Fr(\()p Fn(P)7 b Fr(\),)17 b(deg=1,)p Fn(l)d Fr(=)f(2,)k Fn(C)907 838 y Fm(\030)907 854 y Fr(=)959 851 y Fp(P)998 833 y Fg(1)100 942 y Fn(g)e Fr(=)f(0)f(:)h Fn(L)p Fr(\(0\))g Fm(\032)g Fn(L)p Fr(\()p Fn(P)7 b Fr(\),)17 b(deg=2,)p Fn(l)d Fr(=)f(3,)k Fn(C)907 929 y Fm(\030)907 945 y Fr(=)962 942 y(conic)f(in)h Fp(P)1187 924 y Fg(2)100 1033 y Fn(g)e Fr(=)f(1)f(:)h Fn(L)p Fr(\(0\))g Fm(\032)g Fn(L)p Fr(\(3)p Fn(P)7 b Fr(\),)17 b(deg=3,)p Fn(l)d Fr(=)f(3,)k Fn(C)932 1020 y Fm(\030)932 1035 y Fr(=)987 1033 y(cubic)f(in)g Fp(P)1214 1015 y Fg(2)1239 1033 y Fr(.)100 1143 y(In)i(eac)o(h)h(case,)f(the)h(degree)g(is)f(the)h (degree)f(of)h(the)h(corresp)q(onding)c(curv)o(e,)i(and)g Fn(l)i Fr(is)e(one)h(more)0 1213 y(than)h(the)g(dimension)e(of)j(the)f (corresp)q(onding)e(pro)s(jectiv)o(e)i(space.)f(T)l(o)h(help)g(explain) f(what's)h(going)0 1283 y(on)e(with)g(the)g(geometry)g(here,)f(it)h(is) g(useful)f(to)h(in)o(tro)q(duce)f(the)h(notion)g(of)g(a)g(line)f (bundle.)g(So)h(giv)o(en)0 1353 y Fn(k)r(;)8 b(K)q(;)g(C)20 b Fr(as)c(ab)q(o)o(v)o(e:)100 1463 y Fp(Def)p Fr(:)f(A)i Fp(line)j(bundle)c Fr(on)g Fn(C)j Fr(is)d(a)f(one-dimensional)e(v)o (ector)j(space)g Fm(L)g Fr(o)o(v)o(er)f Fn(K)20 b Fr(together)c(with,)0 1533 y(for)k(ev)o(ery)f(closed)h(p)q(oin)o(t)f Fn(P)26 b Fm(2)20 b Fn(C)t Fr(,)f(a)h(free)f Fn(O)863 1540 y Fk(P)897 1533 y Fr(-mo)q(dule)f Fm(L)1126 1540 y Fk(P)1179 1533 y Fm(\032)h(L)i Fr(of)f(rank)f(1,)h(suc)o(h)e(that)i(for)g(ev)o (ery)0 1603 y Fn(x)14 b Fm(2)g(L)e(\000)f Fr(0,)16 b(one)g(has)g Fn(O)458 1610 y Fk(P)491 1603 y Fn(x)f Fr(=)e Fm(L)620 1610 y Fk(P)670 1603 y Fr(for)k(almost)e(all)i Fn(P)7 b Fr(.)100 1714 y Fp(Example)p Fr(:)16 b Fm(L)f Fr(=)f Fn(K)q(;)8 b Fm(L)549 1721 y Fk(P)598 1714 y Fr(=)14 b Fn(O)689 1721 y Fk(P)722 1714 y Fr(.)j(If)h(w)o(e)e(ha)o(v)o(e)h (just)g(one)g(p)q(oin)o(t,)f(this)h(is)f(the)h(only)g(example)g(of)g(a) 0 1783 y(line)f(bundle,)f(up)h(to)h(isomorphism.)100 1894 y Fp(Def)p Fr(:)d(An)h Fp(isomorphism)d Fr(of)j(line)g(bundles)e Fm(L)p Fn(;)8 b Fm(M)15 b Fr(is)g(a)g Fn(K)t Fr(-linear)e(isomorphism)e (from)k Fm(L)g Fr(to)g Fm(M)0 1964 y Fr(mapping)g Fm(L)239 1971 y Fk(P)289 1964 y Fr(isomorphically)f(on)o(to)i Fm(M)792 1971 y Fk(P)842 1964 y Fr(for)g(all)g Fn(P)7 b Fr(.)100 2074 y(The)15 b(set)h(of)g(isomorphism)d(classes)i(of)h (line)f(bundles)g(forms)f(a)i(group)f(with)g(the)i(tensor)e(pro)q(duct) 0 2144 y(as)22 b(m)o(ultiplication)e([note)i(the)g(analogy)g(with)g(n)o (um)o(b)q(er)e(theory].)h(The)h(unit)g(elemen)o(t)f(is)h(giv)o(en)f(b)o (y)0 2214 y(the)g(trivial)g(line)g(bundle)f(from)h(the)g(ab)q(o)o(v)o (e)g(example,)g(and)f(in)o(v)o(erses)g(are)g(giv)o(en)h(b)o(y)g(the)h Fn(K)j Fr(\(resp.)0 2284 y Fn(O)38 2291 y Fk(P)71 2284 y Fr(\)-linear)16 b(duals.)f(This)h(group)f(is)h(de\014ned)g(to)g(b)q (e)h(the)g Fp(Picard)j(group)15 b Fr(Pic)i Fn(C)i Fr(=)e(Pic)f Fn(K)t Fr(.)100 2394 y(Next)h(time,)f(w)o(e)g(will)g(relate)g(this)g (de\014nition)f(to)i(divisors)d(and)i(giv)o(e)g(some)f(geometric)h(in)o (tuition)0 2464 y(of)h(what)f(this)g(all)h(means.)0 2599
y Fp(Lecture)j(22)c Fr(\(Octob)q(er)h(18,)f(1995\).)0 2670 y(Scrib)q(e:)g(Diane)g(Maclagan)950 2790 y(46)p eop %%Page: 47 47 47 46 bop 100 50 a Fr(Recall)16 b(in)g(the)h(last)f(lecture)g(w)o(e)h (studied)e(\014elds)h(of)h(lo)o(w)f(gen)o(us.)100 120 y(T)l(ak)o(e)f(a)h(divisor)f Fn(D)i Fr(with)f Fn(L)p Fr(\()p Fn(D)q Fr(\))f Fm(6)p Fr(=)f Fm(f)p Fr(0)p Fm(g)p Fr(.)h(This)g(giv)o(es)h(rise)f(to)h(a)g(map)f Fn(\036)e Fr(:)h Fn(C)j Fm(!)d Fp(P)1643 101 y Fk(l)p Fl(\()p Fk(D)q Fl(\))p Fj(\000)p Fl(1)1774 120 y Fr(.)h(This)h(is)0 189 y(not)i(an)f(em)o(b)q(edding)f(in)h(general.)g(The)g(degree)h(in)f (the)h(table)g(in)f(the)h(previous)e(lecture)i(comes)f(from)0 259 y(the)f(degree)f(of)h(the)h(curv)o(e)e(in)g(pro)s(jectiv)o(e)h (space,)f(whic)o(h)g(is)g(w)o(ell)g(de\014ned)g(\(for)h(more)f (information)f(on)0 329 y(this)i(see)h(Hartshorne\).)100 399 y(In)j(Num)o(b)q(er)g(Theory)l(,)f(w)o(e)i(deal)f(with)h(a)f(n)o (um)o(b)q(er)f(\014eld)h Fn(K)25 b Fr(and)20 b(its)g(ring)g(of)h(in)o (tegers)e Fm(O)q Fr(,)i(and)0 468 y(consider)16 b(the)h(class)f(group)g Fn(cl)q Fr(\()p Fm(O)q Fr(\))h(whose)f(elemen)o(ts)h(are)f(the)h (classes)f(of)i(fractional)e(ideals)g(factored)0 538 y(b)o(y)g(the)h(group)e(of)i(principal)e(fractional)h(ideals.)100 608 y(In)g(our)g(con)o(text)h(w)o(e)f(tak)o(e)h(the)g(collection)f(of)h (pro)s(jectiv)o(e)f Fm(O)i Fr(mo)q(dules)d(of)i(rank)f(1.)100 717 y Fp(De\014nition:)35 b Fr(If)16 b Fn(A)h Fr(is)e(a)h(non-zero)e (No)q(etherian)i(ring,)f(a)h(pro)s(jectiv)o(e)f Fn(A)h Fr(mo)q(dule)g(of)g(rank)f(1)h(is)f(a)0 787 y(\014nitely)h(generated)g Fn(A)h Fr(mo)q(dule)f Fn(M)23 b Fr(suc)o(h)15 b(that)i Fm(8)p Fn(p)8 b(M)1025 794 y Fk(p)1061 774 y Fm(\030)1061 790 y Fr(=)1100 794 y Fk(A)1130 799 y Fc(p)1167 787 y Fn(A)1204 794 y Fk(p)1227 787 y Fr(.)100 897 y(A)17 b(one)f (dimensional)e(ob)s(ject)j(is)f(obtained)g(when)g(a)g(pro)s(jectiv)o(e) h Fn(A)g Fr(mo)q(dule)e(is)h(lo)q(calized.)100 1006 y Fp(De\014nition:)45 b Fr(A)20 b(Line)g(Bundle)g(on)g Fn(C)j Fr(is)d(a)g(one)g(dimensional)e Fn(K)24 b Fr(v)o(ector)d(space)e Fm(L)i Fr(together)0 1076 y(with,)15 b(for)g(eac)o(h)g Fn(P)21 b Fm(2)14 b Fn(C)k Fr(a)d(free)h Fm(O)641 1083 y Fk(P)689 1076 y Fr(mo)q(dule)e Fm(L)897 1083 y Fk(P)945 1076 y Fm(\022)f(L)j Fr(of)f(rank)g(1)g(suc)o(h)f(that)i Fm(8)p Fn(x)d Fm(2)h(L)9 b(\000)f(f)p Fr(0)p Fm(g)15 b(8)p Fn(P)21 b Fr(with)0 1146 y(\014nely)16 b(man)o(y)g(exceptions)g Fm(O)552 1153 y Fk(P)585 1146 y Fn(x)e Fr(=)g Fm(L)714 1153 y Fk(P)100 1255 y Fr(The)i(trivial)g(example)g(of)h(a)g(line)f (bundle)f(is)h Fm(L)e Fr(=)g Fn(K)t Fr(,)i(and)g Fm(L)1259 1262 y Fk(P)1306 1255 y Fr(=)e Fm(O)1399 1262 y Fk(P)1448 1255 y Fm(8)p Fn(P)7 b Fr(.)100 1325 y(If)23 b(the)g(curv)o(e)f(has)g (only)g(one)h(p)q(oin)o(t)f(then)g(the)h(trivial)g(line)f(bundle)f(is)i (the)g(only)f(one)h(up)f(to)0 1395 y(isomorphism.)100 1465 y(Another)14 b(example)g(of)g(a)g(line)g(bundle)f(is)h(got)h(b)o (y)f(starting)f(with)i(a)f(divisor)f Fn(D)j Fr(=)1634 1427 y Fh(P)1695 1465 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))p Fn(P)g Fr(,)15 b(and)0 1539 y(taking)i Fm(L)d Fr(=)f Fm(L)p Fr(\()p Fn(D)q Fr(\))j(=)e Fn(K)20 b Fr(and)c Fm(L)630 1546 y Fk(P)677 1539 y Fr(=)e Fm(L)764 1546 y Fk(P)797 1539 y Fr(\()p Fn(D)q Fr(\))i(=)d Fp(m)994 1514 y Fj(\000)p Fk(n)p Fl(\()p Fk(P)5 b Fl(\))994 1554 y Fk(P)100 1609 y Fr(Recall)19 b(that)h(t)o(w)o(o)f(line)g(bundles)f (are)i(isomorphic)d(if)j(there)f(is)h(an)f(isomorphism)d(b)q(et)o(w)o (een)k(the)0 1679 y(v)o(ector)d(spaces)f(taking)h(one)f
Fm(L)582 1686 y Fk(P)633 1679 y Fr(to)h(the)g(other.)f(It)h(is)g(not)f (hard)g(to)h(sho)o(w)f(that)h(ev)o(ery)g(line)f(bundle)g(is)0 1749 y(isomorphic)e(to)k Fm(L)p Fr(\()p Fn(D)q Fr(\))g(for)f(some)f Fn(D)q Fr(.)h(This)f Fn(D)i Fr(is)f(uniquely)f(determined)f(up)i(to)g (a)g(principal)d(divisor.)0 1818 y(So)i(w)o(e)h(ha)o(v)o(e)e(a)i (bijection:)0 1888 y(Pic\()p Fn(C)t Fr(\)=)p Fm(f)p Fr(group)e(of)h (line)h(bundles)d(on)j Fn(C)t Fm(g)834 1880 y Fn(*)834 1890 y(\))897 1888 y Fm(f)g Fr(divisor)e(class)h(group)f(of)i Fn(K)t Fm(g)100 1958 y Fr(where)f([)p Fn(D)q Fr(])e Fm(7!)f(L)p Fr(\()p Fn(D)q Fr(\))19 b(and)d Fm(L)e(7!)732 1921 y Fh(P)793 1958 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))p Fn(P)24 b Fr(where)16 b(the)h Fn(n)p Fr(\()p Fn(P)7 b Fr(\))17 b(are)g(calculated)f(as)g(follo)o(ws:)0 2033 y(Cho)q(ose)f(a)h(basis)f (v)o(ector)g Fn(x)g Fm(2)f(L)9 b(\000)g(f)p Fr(0)p Fm(g)16 b Fr(and)f(lo)q(ok)h(at)g(the)g(submo)q(dule)e(\))1372 2048 y Fk(P)1406 2033 y Fn(x)g Fr(=)f Fp(m)1549 2007 y Fj(\000)p Fk(n)p Fl(\()p Fk(P)5 b Fl(\))1549 2047 y Fk(P)1668 2033 y Fm(L)1702 2040 y Fk(P)1736 2033 y Fr(.)15 b(W)l(e)h(need)0 2103 y(to)j(to)f(sho)o(w)g(that)g Fm(L)p Fr(\()p Fn(D)454 2110 y Fl(1)490 2103 y Fr(+)12 b Fn(D)582 2110 y Fl(2)604 2103 y Fr(\))641 2089 y Fm(\030)641 2105 y Fr(=)696 2103 y Fm(L)p Fr(\()p Fn(D)790 2110 y Fl(1)813 2103 y Fr(\))h Fm(\012)f(L)p Fr(\()p Fn(D)990 2110 y Fl(2)1013 2103 y Fr(\).)19 b(Note)g(that)g(b)o(y)f(abuse)f(of)i (notation)f(this)g(also)0 2172 y(means)d(sho)o(wing)g(this)h(for)h(the) f Fm(L)635 2179 y Fk(P)686 2172 y Fr(as)g(w)o(ell.)100 2242 y(W)l(e)e(de\014ne)g(deg)f Fm(L)i Fr(=)514 2205 y Fh(P)575 2242 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\)deg)p Fn(P)g Fr(.)14 b(This)g(do)q(es)g(not)g(dep)q(end)f(on)h(the)g(basis)f (b)q(ecause)h(deg\()p Fn(f)5 b Fr(\))15 b(=)0 2312 y(0.)100 2382 y(If)h Fn(U)k Fm(\022)13 b Fn(C)20 b Fr(is)c(op)q(en)h(\(i.e.)f (empt)o(y)g(or)g(co\014nite\))h(w)o(e)f(de\014ne)803 2476 y Fm(L)p Fr(\()p Fn(U)5 b Fr(\))16 b(=)999 2429 y Fh(\\)982 2536 y Fk(P)5 b Fj(2)p Fk(U)1079 2476 y Fm(L)1113 2483 y Fk(P)820 2625 y Fm(L)p Fr(\()p Fn(C)t Fr(\))14 b(=)999 2578 y Fh(\\)1011 2685 y Fk(P)1062 2625 y Fm(L)1096 2632 y Fk(P)950 2790 y Fr(47)p eop %%Page: 48 48 48 47 bop 100 50 a Fr(The)16 b(connection)g(with)h(Riemann-Ro)q(c)o(h)d (is)i(that)h(if)f Fm(L)e Fr(=)g Fm(L)p Fr(\()p Fn(D)q Fr(\))k(then)f Fm(L)p Fr(\()p Fn(D)q Fr(\)\()p Fn(C)t Fr(\))e(=)f Fn(L)p Fr(\()p Fn(D)q Fr(\).)0 183 y Fp(Lecture)20 b(23)c Fr(\(Octob)q(er)h(20,)f(1995\).)0 254 y(Scrib)q(e:)g(Emiliano)f (G\023)-25 b(omez)100 388 y(T)l(o)q(da)o(y)18 b(w)o(e)h(will)f(see)h (some)f(examples)g(of)h(line)g(bundles.)e(W)l(e)i(will)f(also)h (de\014ne)f(linear)g(systems)0 458 y(on)f(a)f(curv)o(e)h Fn(C)t Fr(.)f(This)g(will)g(lead)h(us)f(to)i(\014nd)e(a)h(connection)f (b)q(et)o(w)o(een)h(rational)f(maps)f(of)j(curv)o(es)e(in)o(to)0 528 y(pro)s(jectiv)o(e)g(space)g Fp(P)403 510 y Fk(n)431 528 y Fr(\()p Fn(k)r Fr(\))g(and)g(line)g(bundles)f(\(with)i(an)f (additional)g(v)o(ector)g(spa)g(ce)h(structure\).)100 598 y(As)d(usual,)g(let)h Fn(K)q(=k)h Fr(b)q(e)f(a)g(\014nitely)g (generated)f(\014eld)g(extension)h(of)g(trascendence)f(degree)g(1,)h (and)0 668 y(let)i Fn(C)j Fr(b)q(e)d(the)f(\\abstract)g(curv)o(e")g(th) o(us)g(de\014ned:)f Fn(C)20 b Fr(is)c(the)h(set)g(of)f(v)m(aluation)h (rings)e(of)i Fn(K)j Fr(o)o(v)o(er)c Fn(k)r Fr(.)100 739 y(Recall)g(the)h(de\014nition)e(of)i(a)f(line)g(bundle)g(on)g Fn(C)t Fr(:)100 809 y Fp(De\014nition:)e Fr(a)e Fp(line)k(bundle)d Fr(on)g Fn(C)i Fr(is)e(a)f(one-dimensional)e Fn(K)t Fr(-v)o(ector)i
(space)g Fm(L)h Fr(together)f(with,)0 879 y(for)19 b(eac)o(h)f Fn(P)25 b Fm(2)18 b Fn(C)t Fr(,)h(a)g(free)g Fm(O)556 886 y Fk(P)589 879 y Fr(-mo)q(dule)e Fm(L)817 886 y Fk(P)869 879 y Fm(\032)h(L)h Fr(of)g(rank)g(1,)g(suc)o(h)f(that)h(for)g(all)f Fn(x)h Fm(2)f(L)13 b(\000)g(f)p Fr(0)p Fm(g)p Fr(,)18 b(w)o(e)0 949 y(ha)o(v)o(e)e Fm(O)155 956 y Fk(P)199 949 y Fm(\001)11 b Fn(x)j Fr(=)g Fm(L)353 956 y Fk(P)403 949 y Fr(for)i(almost)g(all)g Fn(P)7 b Fr(.)100 1059 y(Also,)24 b(recall)g(that)h(w)o(e)f(de\014ned)g(the)h(set)g(of)g Fp(global)k(sections)c Fr(of)g Fn(C)j Fr(as)c(the)h(set)g Fm(L)p Fr(\()p Fn(C)t Fr(\))i(=)0 1091 y Fh(T)42 1144 y Fk(P)5 b Fj(2)p Fk(C)141 1129 y Fm(L)175 1136 y Fk(P)100 1199 y Fr(W)l(e)20 b(also)g(de\014ned,)f(giv)o(en)g(a)h(divisor)f Fn(D)q Fr(,)i(a)f(line)g(bundle)f Fm(L)p Fr(\()p Fn(D)q Fr(\),)j(for)e(whic)o(h)f(the)h(set)h(of)f(global)0 1269 y(sections)c(is)g Fn(L)p Fr(\()p Fn(D)q Fr(\))i(\(see)f(last)f (lecture\).)100 1382 y(F)l(or)k(the)h(rest)g(of)h(this)f(lecture,)g(w)o (e)g(will)g(assume)e(that)p 1198 1341 28 2 v 22 w Fn(k)k Fr(=)f Fn(k)h Fr(\(i.e.)e(that)h Fn(k)g Fr(is)f(algebraically)0 1452 y(closed\),)16 b(and)g(w)o(e)g(will)g(write)h Fp(P)611 1433 y Fk(n)655 1452 y Fr(for)f(the)h(n)o(th-dimensional)c(pro)s (jectiv)o(e)j(space)g(o)o(v)o(er)g Fn(k)r Fr(.)100 1562 y(No)o(w)g(w)o(e)g(will)h(see)f(some)g(examples)g(of)g(line)g(bundles)f (on)i Fp(P)1243 1544 y Fk(n)1270 1562 y Fr(:)100 1672 y(The)f Fp(tautological)j Fr(line)d(bundle)f(on)h Fp(P)875 1654 y Fk(n)919 1672 y Fr(is)g(de\014ned)g(as)631 1804 y Fm(L)e Fr(=)f Fm(f)p Fr(\()p Fn(x;)8 b(y)r Fr(\))16 b Fm(2)e Fn(k)961 1784 y Fk(n)p Fl([phone] y Fm(\002)d Fp(P)1138 1784 y Fk(n)1165 1804 y Fm(j)p Fn(x)j Fm(2)g Fn(y)r Fm(g)100 1937 y Fr(Here)i(w)o(e)h(are)f(thinking)g(of)g Fp(P)667 1919 y Fk(n)711 1937 y Fr(as)h(the)f(curv)o(e)g(of)h(a)g (function)f(\014eld.)100 2008 y(Note)h(that)g(w)o(e)f(ha)o(v)o(e)g(a)h (natural)e(map)h Fm(L)e(\000)-9 b(!)14 b Fp(P)1024 1990 y Fk(n)1051 2008 y Fr(,)j(giv)o(en)f(b)o(y)g(\()p Fn(x;)8 b(y)r Fr(\))15 b Fm(7\000)-8 b(!)14 b Fn(y)100 2118 y Fr(This)j(line)h(bundle)f(has)h(only)g(one)g(section,)g(namely)f(the)i (zero)f(section.)g(F)l(or)f(this)h(reason,)f(it)i(is)0 2188 y(not)e(useful)e(to)i(us.)100 2258 y(Ho)o(w)o(ev)o(er,)i(things)h (c)o(hange)f(drastically)h(when)g(w)o(e)g(lo)q(ok)h(at)g(its)f(in)o(v)o (erse)f(\(recall)h(that)h(the)g(iso-)0 2328 y(morphism)14 b(classes)h(of)i(line)f(bundles)f(on)h(a)h(curv)o(e)f Fn(C)k Fr(form)15 b(a)i(group)e(Pic)i Fn(C)t Fr(\).)100 2399 y(The)i(in)o(v)o(erse)g(of)h Fm(L)g Fr(is)f(obtained)g(b)o(y)g (dualizing)g(\(see)h(last)f(lecture\).)h(So)g(w)o(e)f(lo)q(ok)h(at)h (the)f(\014b)q(er)0 2468 y(of)g(ev)o(ery)f Fn(y)i Fm(2)d Fp(P)329 2450 y Fk(n)357 2468 y Fr(.)h(This)g(is)f(a)i(\\line",)e (whose)h(in)o(v)o(erse)f(\(dual\))h(giv)o(es)g(us)g(the)g(\014b)q(er)g (o)o(v)o(er)g Fn(y)i Fr(in)e Fm(L)1882 2450 y Fj(\000)p Fl(1)1936 2468 y Fr(.)0 2538 y(Therefore,)d(the)g(in)o(v)o(erse)g(of)g (the)h(tautological)f(line)g(bundle)g(is)581 2670 y Fm(L)615 2649 y Fj(\000)p Fl(1)683 2670 y Fr(=)d Fm(f)p Fr(\()p Fn(x;)8 b(y)r Fr(\))p Fm(j)p Fn(y)18 b Fm(2)c Fp(P)1016 2649 y Fk(n)1043 2670 y Fn(;)8 b(x)14 b Fm(2)g Fn(k)1182 2649 y Fk(n)p Fl([phone] y Fn(=y)1310 2649 y Fj(?)1344 2670 y Fm(g)950 2790 y Fr(48)p eop %%Page: 49 49 49 48 bop 0 50 a Fr(Here)24 b Fn(k)153 32 y Fk(n)p Fl(+1)230 50 y Fn(=y)281 32 y Fj(?)341 50 y Fr(=)i(Hom)510 57 y Fk(k)534 50 y Fr(\()p Fn(y)r(;)8 b(k)r Fr(\).)25 b(W)l(e)f(ha)o(v)o(e)f
(an)h(inner)e(pro)q(duct)i Fn(k)1336 32 y Fk(n)p Fl(+1)1429 50 y Fm(\002)16 b Fn(k)1512 32 y Fk(n)p Fl(+1)1615 50 y Fm(\000)-9 b(!)26 b Fn(k)r Fr(,)d(sending)0 120 y(\()p Fn(x;)8 b(y)r Fr(\))15 b Fm(7\000)-8 b(!)13 b(h)p Fn(x;)8 b(y)r Fm(i)p Fr(,)18 b(and)e Fn(y)492 101 y Fj(?)540 120 y Fr(=)e Fm(f)p Fn(x)p Fm(jh)p Fn(x;)8 b(y)r Fm(i)15 b Fr(=)f(0)p Fm(g)100 189 y Fr(Note)j(again)f(that)h(there)f(is)g(a)h (natural)e(map)h Fm(L)999 171 y Fj(\000)p Fl(1)1067 189 y Fm(\000)-9 b(!)14 b Fp(P)1200 171 y Fk(n)1227 189 y Fr(.)100 299 y(It)19 b(turns)f(out)h(that)g(Pic)g Fp(P)618 281 y Fk(n)664 299 y Fr(is)g(in\014nite)f(cyclic,)h(with)g(generators)e Fm(L)j Fr(and)e Fm(L)1587 281 y Fj(\000)p Fl(1)1641 299 y Fr(.)h(The)f(latter)i(is)0 369 y(the)d(\\b)q(etter")h(generator,)e(b) q(ecause)h(it)g(has)g(lots)f(of)i(sections,)e(as)h(opp)q(osed)f(to)h (the)h(tautological)f(line)0 438 y(bundle.)100 548 y(Next)g(let)g(us)f (consider)f(line)h(bundles)f(on)i Fp(P)937 530 y Fl(1)959 548 y Fr(:)100 658 y(Recall)d(that)h(Pic)g Fn(C)j Fr(is)d(isomorphic)d (to)j(the)g(divisor)f(class)g(group.)g(If)h(the)g(gen)o(us)e(of)i(the)g (function)0 727 y(\014eld)g(is)h(zero,)g(then)g(the)g(degree)f(map)h (Pic)g Fn(C)j Fm(\000)-9 b(!)14 b Fp(Z)i Fr(is)g(injectiv)o(e)g (\(exercise)g(19)g(on)g(the)g(homew)o(ork\).)0 799 y(With)k(our)e (assumption)g(that)p 601 758 28 2 v 20 w Fn(k)i Fr(=)e Fn(k)r Fr(,)h(w)o(e)h(actu)f(ally)h(ha)o(v)o(e)f(an)g(isomorphism)o(.)e (So)i(w)o(e)g(can)h(write)f(a)0 869 y(line)d(bundle)f(for)i(eac)o(h)f (in)o(teger.)100 978 y(W)l(e)g(can)h(form)e(our)h(function)g(\014eld)g (as)g Fn(K)i Fr(=)c Fn(k)r Fr(\()p Fn(t)p Fr(\))j(=)f(degree-0-piece)f (of)i Fn(S)1528 960 y Fj(\000)p Fl(1)1581 978 y Fn(k)r Fr([)p Fn(x)1651 985 y Fl(0)1673 978 y Fn(;)8 b(x)1723 985 y Fl(1)1746 978 y Fr(],)16 b(where)100 1048 y Fn(t)d Fr(=)190 1027 y Fk(x)213 1032 y Fd(1)p 190 1036 43 2 v 190 1065 a Fk(x)213 1070 y Fd(0)238 1048 y Fr(,)j(and)g Fn(S)j Fr(is)d(the)h(set)g(of)g(nonzero)e(homogeneous)g(elemen)o(ts)h (of)g Fn(k)r Fr([)p Fn(x)1506 1055 y Fl(0)1529 1048 y Fn(;)8 b(x)1579 1055 y Fl(1)1601 1048 y Fr(].)17 b(W)l(e)g(can)f(write) 733 1193 y Fn(S)767 1172 y Fj(\000)p Fl(1)820 1193 y Fn(k)r Fr([)p Fn(x)890 1200 y Fl(0)912 1193 y Fn(;)8 b(x)962 1200 y Fl(1)985 1193 y Fr(])13 b(=)1067 1145 y Fh(M)1065 1252 y Fk(n)p Fj(2)p Fg(Z)1152 1193 y Fn(R)1190 1200 y Fk(n)0 1365 y Fr(The)j(piece)h(of)g(degree)f(zero)g(is)g Fn(R)632 1372 y Fl(0)668 1365 y Fr(=)e Fn(K)20 b Fr(What)d(ab)q(out)f (the)h(other)f(pieces?)100 1435 y(They)e(are)h(isomorphic)d(to)j(one)f (another)g(up)q(on)g(m)o(ultiplication)f(b)o(y)h(a)h(suitable)f(p)q(o)o (w)o(er)f(of)i Fn(x)1846 1442 y Fl(0)1869 1435 y Fr(.)f(So)0 1504 y(dim)84 1511 y Fk(K)121 1504 y Fn(R)159 1511 y Fk(n)200 1504 y Fr(=)g(1.)i(Th)o(us)f(they)i(are)g(one-dimensional)c(v) o(ector)k(spaces)e(o)o(v)o(er)h Fn(K)t Fr(.)100 1574 y(Eac)o(h)f Fn(R)261 1581 y Fk(n)305 1574 y Fr(has)h(a)g(line)g(bundle) g(structure)f(of)i(degree)f(n.)100 1644 y(W)l(e)g(ha)o(v)o(e)f(that)h Fn(R)444 1651 y Fl(1)482 1644 y Fr(is)f(our)g(\\nice")h(generator)f(of) h(Pic)g Fp(P)1173 1626 y Fl(1)1195 1644 y Fr(,)g(ha)o(ving)f(degree)g (1.)h(The)f(tautological)0 1714 y(bundle)f(on)g Fp(P)268 1696 y Fl(1)306 1714 y Fr(has)g(degree)h(-1,)f(and)h(so,)f(as)h(remark) o(ed)e(ab)q(o)o(v)o(e,)h(it)i(has)e(no)h(sections)f(other)h(than)f (zero,)0 1783 y(b)q(ecause)i(it)h(corresp)q(onds)e(to)h(a)h(divisor)e Fn(D)j Fr(of)f(degree)f(-1,)g(and)g(so)g Fm(L)p Fr(\()p Fn(C)t Fr(\))e(=)g Fn(L)p Fr(\()p Fn(D)q Fr(\))h(=)f(0.)100 1853 y(If)21 b(w)o(e)f(giv)o(e)h Fn(R)376 1860 y Fl(1)419 1853 y Fr(a)f(line)h(bundle)e(structure)h(and)g(call)h(it)g Fn(R)1237 1860 y Fl(1)1279 1853 y Fr(also,)f(then)h Fn(x)1543
1860 y Fl(0)1566 1853 y Fn(;)8 b(x)1616 1860 y Fl(1)1660 1853 y Fm(2)21 b Fn(R)1752 1860 y Fl(1)1774 1853 y Fr(\()p Fn(C)t Fr(\),)f(i.e.)0 1923 y Fn(x)28 1930 y Fl(0)51 1923 y Fn(;)8 b(x)101 1930 y Fl(1)140 1923 y Fr(are)17 b(global)e(sections.)100 2032 y(Ha)o(ving)i(seen)g(these)h(examples,)f (let)h(us)f(again)g(fo)q(cus)h(on)f(\014nding)g(a)g(nice)h(connection)f (b)q(et)o(w)o(een)0 2102 y(morphisms)d(from)h(a)i(curv)o(e)f(to)h(pro)s (jectiv)o(e)f(space)g(and)g(line)g(bundles)f(on)h(the)h(curv)o(e.)100 2172 y(W)l(e)25 b(will)f(see)h(that)g(this)g(connection)f(is)g (realized)h(through)e(some)h(sp)q(ecial)h(sets)g(of)g(divisors)0 2242 y(\(basep)q(oin)o(t)16 b(free)g(linear)g(systems)g(on)g(the)h (curv)o(e\),)f(whic)o(h)f(w)o(e)i(are)f(ab)q(out)g(to)h(de\014ne.)100 2311 y(Also,)i(w)o(e)h(will)f(end)h(up)f(with)h(an)g(additional)e(v)o (ector)i(space)g(structure)f(on)g(the)h(line)g(bundles,)0 2381 y(and)d(w)o(e)g(will)f(often)i(b)q(e)f(able)g(to)h(use)f(the)g (Riemann-Ro)q(c)o(h)e(theorem)h(to)i(obtain)e(information)g(ab)q(out)0 2451 y(this)g(additional)f(structure.)100 2560 y(W)l(e)h(are)h(still)f (assuming)e(that)p 690 2520 28 2 v 17 w Fn(k)h Fr(=)f Fn(k)r Fr(,)i(to)h(sa)o(v)o(e)f(us)g(some)f(trouble.)100 2670 y(Let)i Fn(K)q(;)8 b(k)18 b Fr(and)e Fn(C)k Fr(b)q(e)d(as)f (usual.)950 2790 y(49)p eop %%Page: 50 50 50 49 bop 100 50 a Fp(De\014nition:)16 b Fr(a)f Fp(linear)k(system)14 b Fr(on)h Fn(C)k Fr(is)c(a)g(set)g Fn(S)j Fr(of)d(e\013ectiv)o(e)i (divisors)c(of)j Fn(C)i Fr(s.t.)d Fm(9)g Fr(a)g(divisor)0 120 y Fn(D)42 101 y Fj(0)73 120 y Fr(of)i Fn(C)j Fr(and)c(a)g(sub-)p Fn(k)r Fr(-v)o(ector)e(space)i Fn(V)25 b Fm(\032)14 b Fn(L)p Fr(\()p Fn(D)944 101 y Fj(0)959 120 y Fr(\))j(of)g(dimension)d (n+1)i(s.t.)667 267 y Fn(S)g Fr(=)e Fm(f)p Fr(\()p Fn(f)5 b Fr(\))12 b(+)f Fn(D)963 247 y Fj(0)978 267 y Fm(j)p Fn(f)19 b Fm(2)14 b Fn(V)22 b Fm(\000)11 b(f)p Fr(0)p Fm(gg)100 421 y Fp(De\014nition:)18 b Fr(Let)f Fn(S)j Fr(b)q(e)d(a)g(linear)e(system)i(on)f Fn(C)t Fr(.)g(W)l(e)h(sa)o(y)g (that)g Fn(P)k Fm(2)14 b Fn(C)21 b Fr(is)16 b(a)h Fp(basep)r(oin)n(t)g Fr(of)g Fn(S)0 491 y Fr(if)g Fm(8)p Fn(D)d Fm(2)g Fn(S;)8 b(D)15 b Fm(\025)f Fn(P)7 b Fr(.)100 567 y(W)l(e)16 b(sa)o(y)g(that)h Fn(S)i Fr(is)e Fp(basep)r(oin)n(t)i(free)d Fr(if)g(it)h(has)f(no)g (basep)q(oin)o(ts.)100 683 y(Remark:)f(Note)i(that)g(it)g(is)f(not)h (required)e(that)i(a)g(basep)q(oin)o(t)e(of)i Fn(S)i Fr(b)q(e)e(an)f(elemen)o(t)g(of)h Fn(S)s Fr(.)100 798 y(W)l(e)22 b(ha)o(v)o(e)f(an)g(equiv)m(alence)h(b)q(et)o(w)o(een,)f(on) h(one)f(hand,)g(non-degenerate)f(morphisms)f(from)i Fn(C)0 868 y Fr(to)f Fp(P)103 850 y Fk(n)130 868 y Fr(,)f(up)g(to)h(Aut)g Fp(P)445 850 y Fk(n)491 868 y Fr(\(i.e.)g(up)f(to)g(linear)g(c)o (hanges)f(of)h(co)q(ordinates\),)g(and,)g(on)g(the)g(other)g(hand,)0 938 y(basep)q(oin)o(t)c(free)i(linear)f(systems)f(on)i Fn(C)t Fr(.)100 1014 y(The)f(corresp)q(ondence)f(from)h(the)g(\014rst)g (set)h(to)g(the)g(second)e(is)i(obtained)e(as)h(follo)o(ws:)100 1089 y(Giv)o(en)i Fn(C)t Fr(,)h(it)g(will)g(in)o(tersect)f(an)h (\(n-1\)-dimensional)d(h)o(yp)q(erplane)i(of)h Fp(P)1501 1071 y Fk(n)1548 1089 y Fr(at)g(a)g(\014nite)g(n)o(um)o(b)q(er)0 1159 y(of)g(p)q(oin)o(ts)g(\(coun)o(t)g(m)o(ultiplicities\),)f(th)o(us) g(giving)h(rise)f(to)i(an)f(e\013ectiv)o(e)h(divisor.)e(If)h(w)o(e)h (consider)d(all)0 1229 y(suc)o(h)e(h)o(yp)q(erplanes,)f(w)o(e)i(obtain) f(a)h(set)g Fn(S)j Fr(of)d(e\013ectiv)o(e)h(divisors,)d(whic)h(h)h (will)g(b)q(e)g(our)f(basep)q(oin)o(t)g(free)0 1299 y(linear)h(system)g (on)g Fn(C)t Fr(.)100 1414 y(Next)e(time)g(w)o(e)f(will)g(\014nally)g (obtain)g(the)h(connection)f(b)q(et)o(w)o(een)g(the)h(ab)q(o)o(v)o(e)f (morphisms)e(and)i(line)0 1484 y(bundles)i(on)g Fn(C)t
Fr(,)h(with)g(a)g(certain)g(additional)f(structure.)g(W)l(e)h(will)g (th)o(us)g(get)g(another)g(set)g(\(in)o(v)o(olving)0 1554 y(line)g(bundles\))f(whic)o(h)h(giv)o(es)g(us)g(equiv)m(alences)g (to)h(the)g(ab)q(o)o(v)o(e)f(t)o(w)o(o)g(sets.)0 1714 y Fp(Lecture)k(24)c Fr(\(Octob)q(er)h(23,)f(1995\).)0 1790 y(Scrib)q(e:)g(La)o(wren)f(Smitline)0 1950 y(T)l(o)q(da)o(y)l(,)e (Professor)e(Lenstra)i(w)o(ore)f(a)i(plaid)e(jac)o(k)o(et,)h(strip)q (ed)g(shirt,)f(and)h(a)g(red)f(and)h(na)o(vy)g(blue)g(strip)q(ed)0 2020 y(tie)k(with)f(a)h(narro)o(w)e(gold)h(strip)q(e.)100 2101 y(Recall)g Fn(C)k Fr(is)c(a)g(curv)o(e)g(corresp)q(onding)e(to)j (a)g(function)f(\014eld)g Fn(K)k Fr(o)o(v)o(er)c Fn(k)r Fr(.)0 2217 y(A)i Fq(line)m(ar)i(system)e Fr(on)f Fn(C)k Fr(is)c(a)h(set)g Fn(S)i Fr(of)e(e\013ectiv)o(e)h(divisors)d(on)h Fn(C)k Fr(for)d(whic)o(h)e(there)i(exists)g(a)f(divisor)0 2287 y Fn(D)h Fr(on)e Fn(C)k Fr(and)c(a)h(subspace)e Fn(V)25 b Fm(\032)13 b Fn(L)p Fr(\()p Fn(D)q Fr(\))18 b(suc)o(h)d(that)0 2403 y Fn(S)h Fr(=)e Fm(f)p Fr(\()p Fn(f)5 b Fr(\))12 b(+)f Fn(D)16 b Fr(:)d Fn(f)20 b Fm(2)14 b Fn(V)22 b Fm(\000)11 b(f)p Fr(0)p Fm(gg)p Fr(.)0 2478 y(\(Note)18 b(that)f(the)f(divisors)f(\()p Fn(f)5 b Fr(\))19 b(are)d(elemen)o(ts)f(of)i Fp(P)p Fn(V)1037 2465 y Fm(\030)1037 2481 y Fr(=)1090 2478 y(\()p Fn(V)22 b Fm(\000)11 b(f)p Fr(0)p Fm(g)p Fr(\))p Fn(=k)1357 2460 y Fj(\002)1390 2478 y Fr(.\))0 2594 y(A)17 b(p)q(oin)o(t)f Fn(P)21 b Fm(2)14 b Fn(C)20 b Fr(is)c(a)g Fq(b)m(ase)j(p)m(oint)f Fr(for)e Fn(S)j Fr(if)e(for)f(all)g Fn(D)g Fm(\032)d Fn(S;)j(D)g Fm(\025)d Fn(P)7 b Fr(.)0 2670 y Fn(S)19 b Fr(is)d(called)g Fq(b)m(ase)j(p)m(oint)g(fr)m(e)m(e)f Fr(if)e(it)h(has)f(no)g(base)g(p)q(oin)o(ts.)950 2790 y(50)p eop %%Page: 51 51 51 50 bop 0 50 a Fr(Example)16 b(0.)g Fn(S)g Fr(=)e Fm(;)p Fr(.)i(T)l(ak)o(e)h(an)o(y)f Fn(D)674 32 y Fj(0)705 50 y Fr(and)g Fn(V)25 b Fr(=)13 b Fm(f)p Fr(0)p Fm(g)p Fr(.)k(Ev)o(ery)f (p)q(oin)o(t)g(is)g(a)h(base)f(p)q(oin)o(t.)0 162 y(Example)h(1.)g(T)l (ak)o(e)g(an)o(y)g Fn(D)522 144 y Fj(0)554 162 y Fr(and)f Fn(V)27 b Fr(=)14 b Fn(L)p Fr(\()p Fn(D)q Fr(\).)19 b(The)e(resulting)f (linear)g(system)h(is)g(called)g(a)g(complete)0 231 y(linear)f(system)g (and)g(is)g(denoted)g Fm(j)p Fn(D)695 213 y Fj(0)709 231 y Fm(j)p Fr(.)0 304 y Fm(j)p Fn(D)56 285 y Fj(0)70 304 y Fm(j)e Fr(=)g Fm(f)p Fn(D)h Fm(\025)f Fr(0)f(:)h Fn(D)i Fm(\030)d Fn(D)502 285 y Fj(0)517 304 y Fm(g)p Fn(:)0 376 y(P)24 b Fr(is)16 b(a)g(base)g(p)q(oin)o(t)g(for)g Fm(j)p Fn(D)520 358 y Fj(0)535 376 y Fm(j)g Fr(if)h(and)f(only)g(if)h Fn(l)q Fr(\()p Fn(D)940 358 y Fj(0)966 376 y Fm(\000)10 b Fn(P)d Fr(\))15 b(=)e Fn(l)q Fr(\()p Fn(D)1217 358 y Fj(0)1232 376 y Fr(\).)0 517 y(Supp)q(ose)i Fn(S)j Fr(is)e(a)g(nonempt)o(y)f(linear)g(system.)h(Then)f(\()p Fn(D)1074 499 y Fj(0)1089 517 y Fn(;)8 b(V)j Fr(\))17 b(is)f(uniquely)f(determined)g(up)g(to)i Fn(D)1872 499 y Fj(0)1900 517 y Fm(!)0 587 y Fn(D)42 569 y Fj(0)68 587 y Fr(+)10 b(\()p Fn(g)r Fr(\),)17 b Fn(V)25 b Fm(!)14 b Fn(g)356 569 y Fj(\000)p Fl(1)409 587 y Fn(V)d Fr(,)16 b(for)h Fn(g)e Fm(2)f Fn(K)688 569 y Fj(\002)722 587 y Fr(.)0 659 y(If)j Fn(D)92 641 y Fj(0)123 659 y Fr(and)g Fn(D)263 641 y Fj(00)306 659 y Fr(giv)o(e)f(the)h(same)f Fn(S)s Fr(,)g(then)h(they)g(are)g(linearly)f(equiv)m(alen)o(t.)g(So)h (only)f(the)h(divisor)f(class)0 728 y(of)21 b Fn(D)103 710 y Fj(0)139 728 y Fr(is)g(determined)e(up)i(to)g(equiv)m(alence.)h
(So)e(only)h(the)h(line)e(bundle)g(asso)q(ciated)h(to)g Fn(D)1775 710 y Fj(0)1811 728 y Fr(is)g(w)o(ell)0 798 y(de\014ned)16 b(in)g(terms)f(of)i Fn(S)s Fr(.)0 910 y(What)g(do)q(es)f(this)g Fn(V)28 b Fr(ha)o(v)o(e)16 b(to)h(do)f(with)g(the)h(linear)f(system?)0 982 y(A)h(nonempt)o(y)f (linear)g(system)h(corresp)q(onds)e(to)i(a)g(p)q(ear)g(\(of)h(the)f(b)q (osc)g(v)m(ariet)o(y\))h(consisting)d(of)j(a)f(line)0 1052 y(bundle)e Fm(L)i Fr(plus)f(a)g(subspace)f Fn(V)25 b Fm(\032)14 b(L)p Fr(\()p Fn(C)t Fr(\))j(of)f(the)h(space)f(of)h (global)f(sections)g(of)g Fm(L)h Fr(\(=)1649 1015 y Fh(T)1691 1067 y Fk(P)5 b Fj(2)p Fk(C)1790 1052 y Fm(L)1824 1059 y Fk(P)1858 1052 y Fn(:)p Fr(\))0 1164 y Fn(P)24 b Fr(is)16 b(a)g(base)g(p)q(oin)o(t)g Fm(,)e Fn(V)25 b Fm(\032)13 b Ff(m)596 1171 y Fk(P)630 1164 y Fm(L)664 1171 y Fk(P)697 1164 y Fr(.)0 1236 y(base)j(p)q(oin)o(t)g(free)h Fm(,)f Fr(the)h(global)f(sections)g(in)g Fn(V)27 b Fr(generate)17 b Fm(L)p Fn(:)0 1308 y Fr(Y)l(ou)f(are)h(supp)q(osed)e(to)h(b)q(e)h (able)f(to)h(do)f(that)h(without)g(thinking)f(and)g(hop)q(efully)g (will)g(do)g(that.)0 1420 y(No)o(w)g(w)o(e)h(ha)o(v)o(e)f(three)g (equiv)m(alen)o(t)h(form)o(ulations,)d(dep)q(ending)h(on)h(y)o(our)g (language.)0 1492 y Fm(f)p Fr(nondegenerate)f(maps)h Fn(C)h Fm(!)c Fp(P)640 1474 y Fk(n)668 1492 y Fm(g)p Fn(=)731 1479 y Fm(\030)731 1495 y Fr(=)770 1499 y Fg(P)801 1490 y Fc(n)828 1492 y Fm($)0 1564 y(f)p Fr(base)j(p)q(oin)o(t)g(free)h (linear)e(systems)h(on)g Fn(C)k Fr(where)c(dim)g Fn(V)25 b Fr(=)13 b Fn(n)e Fr(+)g(1)p Fm(g)j($)0 1636 y(f)p Fr(line)i(bundles)f Fm(L)i Fr(plus)e Fn(V)28 b Fr(a)17 b(dimension)d Fn(n)d Fr(+)g(1)16 b(space)g(of)h(global)f(sections)g(of)h Fm(L)g Fr(generating)e Fm(Lg)p Fr(.)100 1780 y(Let)22 b Fn(f)29 b Fr(:)23 b Fn(C)j Fm(!)d Fp(P)458 1762 y Fk(n)485 1780 y Fr(,)f(with)g(image)f Fn(C)829 1762 y Fj(0)842 1780 y Fr(.)h(Let)h Fn(H)k Fm(\032)c Fp(P)1142 1762 y Fk(n)1191 1780 y Fr(b)q(e)f(a)g(h)o(yp)q(erplane.)f Fn(H)e Fm(\\)c Fn(C)1735 1762 y Fj(0)1770 1780 y Fr(giv)o(es)21 b(an)0 1850 y(e\013ectiv)o(e)e(divisor)d(on)i Fn(C)t Fr(,)f(and)g(the)h(set)g Fm(f)p Fn(H)f Fm(\\)12 b Fn(C)931 1831 y Fj(0)960 1850 y Fr(:)18 b(h)o(yp)q(erplane)e Fn(H)21 b Fm(\032)16 b Fp(P)1405 1831 y Fk(n)1432 1850 y Fm(g)i Fr(has)f(the)h(structure)f(of) h(a)0 1919 y(pro)s(jectiv)o(e)g(space,)g(and)g(is)h(the)f(same)g(as)h Fn(S)s Fr(.)f(If)h Fm(M)f Fr(is)g(the)h(tautological)f(line)h(bundle)e (on)h Fp(P)1793 1901 y Fk(n)1821 1919 y Fr(,)g(then)0 1989 y Fn(S)e Fr(=)e Fn(f)129 1971 y Fj(\003)152 1989 y Fm(M)p Fr(.)0 2101 y(Supp)q(ose)g Fn(C)19 b Fr(is)c(a)h(curv)o(e)f (and)g(one)g(has)g(an)g Fn(n)9 b Fr(+)g(1)16 b(dimensional)d(space)i (of)h(global)e(sections.)h(Then)g(one)0 2171 y(has)h Fn(f)113 2178 y Fl(0)144 2171 y Fn(:)8 b(:)g(:)g(f)234 2178 y Fk(n)278 2171 y Fr(not)17 b(all)f(0)h(at)g(an)o(y)f(p)q(oin)o (t,)g(b)o(y)g(base)g(p)q(oin)o(t)g(freeness.)f(So)i(let)0 2243 y Fn(f)j Fr(:)13 b Fn(P)21 b Fm(7!)14 b Fr(\()p Fn(f)231 2250 y Fl(0)254 2243 y Fr(\()p Fn(P)7 b Fr(\))14 b(:)g Fn(:)8 b(:)g(:)14 b Fr(:)g Fn(f)497 2250 y Fk(n)524 2243 y Fr(\()p Fn(P)7 b Fr(\)\))15 b Fm(2)f Fp(P)721 2225 y Fk(n)748 2243 y Fr(.)100 2315 y(When)h(w)o(e)g(ha)o(v)o(e)g Fn(f)k Fr(:)14 b Fn(C)j Fm(!)c Fp(P)656 2297 y Fk(n)684 2315 y Fr(,)i(w)o(e)g(ha)o(v)o(e)g Fn(C)j Fr(as)d(an)g(algebraic)g (subset.)f(Ho)o(w)h(do)g(w)o(e)g(express)g(the)0 2385 y(condition)h(that)h Fn(C)i Fr(is)d(a)h(closed)f(em)o(b)q(edding?)0 2457 y(A)h(v)o(ery)f(ample)g(line)g(bundle)f(is)i(generated)f(b)o(y)g (its)g(global)g(sections....)0 2598 y Fp(Lecture)k(25)c
Fr(\(Octob)q(er)h(25,)f(1995\).)0 2670 y(Scrib)q(e:)g(Emiliano)f(G\023) -25 b(omez)950 2790 y(51)p eop %%Page: 52 52 52 51 bop 100 50 a Fr(Let)17 b Fn(k)e Fm(\032)f Fn(K)20 b Fr(and)c Fn(C)k Fr(b)q(e)d(as)f(usual,)f(with)i Fn(k)h Fr(algebraically)d(closed.)100 125 y(Recall)i(that)g(last)g(time)h(w)o (e)f(got)g(a)h(\\dictionary")e(\(an)i(equiv)m(alence\))g(b)q(et)o(w)o (een)f(the)g(set)h(of)f(mor-)0 195 y(phisms)100 271 y Fn(')c Fr(:)h Fn(C)j Fm(\000)-9 b(!)14 b Fp(P)360 253 y Fk(n)404 271 y Fr(\(non-degenerate,)h(and)h(up)g(to)h(linear)e(c)o (hanges)g(of)i(co)q(ordinates)f(of)h Fp(P)1753 253 y Fk(n)1780 271 y Fr(\))100 346 y(and)e(the)i(set)g(of)g(pairs)100 422 y(\()p Fm(L)p Fn(;)8 b(V)k Fr(\),)19 b(where)g Fm(L)h Fr(is)f(a)g(line)g(bundle)f(on)h Fn(C)k Fr(and)18 b Fn(V)30 b Fm(\032)18 b(L)p Fr(\()p Fn(C)t Fr(\))i(is)f(an)g(n+1-dimensional)d (v)o(ector)0 492 y(space)g(of)h(global)f(sections)f(of)i Fm(L)g Fr(suc)o(h)f(that)g(\\)p Fn(V)28 b Fr(generates)16 b Fm(L)p Fr(.")100 567 y(\(W)l(e)g(will)f(not)g(de\014ne)g(what)g(it)h (means,)e(in)h(general,)f(for)h(the)h(global)e(sections)h(in)g Fn(V)27 b Fr(to)15 b(generate)0 637 y Fm(L)p Fr(.)i(See)f(Hartshorne,)f (Ch.2,)h(sec.7\))100 752 y(A)h(question)h(that)f(comes)g(up)g(is:)g (whic)o(h)f(pairs)g(\()p Fm(L)p Fn(;)8 b(V)k Fr(\))18 b(giv)o(e)f(rise)g(\(under)g(this)g(equiv)m(alence\))h(to)0 830 y(closed)e(em)o(b)q(eddings)e(\(i.e.)j Fn(')c Fr(:)h Fn(C)656 802 y Fj(')632 830 y Fm(\000)-9 b(!)p Fn(')p Fr(\()p Fn(C)t Fr(\)\))16 b(?)100 906 y(W)l(e)h(will)g(not)g(answ)o(er) f(this)h(question)f(in)h(general.)f(Let)i(us)f(consider)f(only)g(the)i (case)f Fn(V)26 b Fr(=)15 b Fm(L)p Fr(\()p Fn(C)t Fr(\))0 976 y(\(i.e.)i(the)f(space)h(of)f(all)g(global)g(sections\):)100 1051 y(Recall)21 b(that)g(an)o(y)g(line)g(bundle)f Fm(L)i Fr(is)f(isomorphic)e(to)j Fm(L)p Fr(\()p Fn(D)q Fr(\))h(\(the)f(bundle) e(coming)g(from)h(the)0 1121 y(divisor)15 b Fn(D)q Fr(,)i(de\014ned)f (some)g(lectures)g(ago\))h(for)f(some)g(divisor)f Fn(D)q Fr(.)100 1196 y(So)h(let)h Fn(D)h Fr(b)q(e)f(suc)o(h)e(that)i(our)f (line)g(bundle)f Fm(L)i Fr(is)f(isomp)q(orphic)f(to)h Fm(L)p Fr(\()p Fn(D)q Fr(\).)100 1272 y(Let)j Fn(S)j Fr(b)q(e)e(the)f(complete)g(linear)f(system)h(asso)q(ciated)g(to)g Fn(D)q Fr(.)h(W)l(e)f(sa)o(w)g(last)g(time)g(that)g Fn(P)26 b Fr(is)19 b(a)0 1342 y(basep)q(oin)o(t)c(for)h Fn(S)i Fr(i\013)e Fn(l)q Fr(\()p Fn(D)c Fm(\000)e Fn(P)d Fr(\))14 b(=)g Fn(l)q Fr(\()p Fn(D)q Fr(\).)j(And)f(w)o(e)f(also)h(remark)o(ed)e (that)j Fn(S)h Fr(is)e(basep)q(oin)o(t)f(free)h(i\013)g(the)0 1412 y(global)g(sections)g(of)g Fn(V)28 b Fr(generate)16 b Fm(L)p Fr(.)100 1487 y(vCom)o(bining)e(these)j(t)o(w)o(o)f(results,)f (w)o(e)h(see)h(that)100 1563 y Fn(V)25 b Fr(=)13 b Fm(L)p Fr(\()p Fn(C)t Fr(\))k(generates)f Fm(L)h Fr(i\013)g Fm(8)p Fn(P)i Fr(:)14 b Fn(l)q Fr(\()p Fn(D)f Fm(\000)d Fn(P)d Fr(\))14 b Fn(<)g(l)q Fr(\()p Fn(D)q Fr(\).)100 1641 y(Since)j(w)o(e)h(are)g(assuming)e(that)p 718 1600 28 2 v 18 w Fn(k)i Fr(=)e Fn(k)k Fr(\(so)e(deg)p Fn(P)24 b Fr(=)18 b(1)g(for)g(all)g Fn(P)23 b Fm(2)16 b Fn(C)22 b Fr(closed\),)17 b(the)i(Riemann)0 1710 y(theorem)d(tells)g(us)g(that) h Fn(l)q Fr(\()p Fn(D)c Fm(\000)e Fn(P)c Fr(\))17 b(and)f Fn(l)q Fr(\()p Fn(D)q Fr(\))h(ma)o(y)f(di\013er)g(b)o(y)g(at)h(most)f (1.)100 1786 y(Therefore,)f(the)i(ab)q(o)o(v)o(e)f(inequalit)o(y)g(is)g (equiv)m(alen)o(t)h(to)g Fn(l)q Fr(\()p Fn(D)c Fm(\000)e Fn(P)c Fr(\))14 b(=)f Fn(l)q Fr(\()p Fn(D)q Fr(\))g Fm(\000)d Fr(1.)100 1861 y(F)l(rom)15 b(this)h(discussion)e(w)o(e)i(get)h(the)g (follo)o(wing)e(result:)100 1937 y(\()p Fm(L)p Fn(;)8
b(V)k Fr(\))19 b(with)h Fn(V)29 b Fr(=)18 b Fm(L)p Fr(\()p Fn(C)t Fr(\))i(giv)o(es)f(rise)f(to)i(a)f(closed)g(em)o(b)q(edding)f (if)h(and)g(only)g(if)g Fm(8)p Fn(P)q(;)8 b(Q)19 b Fr(\(closed)0 2007 y(p)q(oin)o(ts)d(of)g Fn(C)t Fr(\),)g(w)o(e)h(ha)o(v)o(e)f Fn(l)q Fr(\()p Fn(D)d Fm(\000)d Fn(P)18 b Fm(\000)11 b Fn(Q)p Fr(\))k(=)e Fn(l)q Fr(\()p Fn(D)q Fr(\))g Fm(\000)d Fr(2.)100 2122 y(Note)15 b(that)f(w)o(e)g(can)g(use)f(the)i(Riemann-Ro) q(c)o(h)c(Theorem)i(to)h(c)o(hec)o(k)g(whether)g(the)g(last)g (condition)0 2192 y(holds)e(or)g(not.)h(Also)g(note)g(that)g(for)g Fn(P)20 b Fm(6)p Fr(=)14 b Fn(Q)p Fr(,)f(this)g(condition)f(merely)g (guaran)o(tees)g(that)h Fn(')f Fr(is)h(injectiv)o(e.)0 2262 y(But)k(when)f(w)o(e)g(consider)f(the)i(case)g Fn(P)j Fr(=)14 b Fn(Q)p Fr(,)j(w)o(e)f(get)h(that)g Fn(')f Fr(is)g(a)h(c)g (losed)e(em)o(b)q(edding.)100 2377 y(F)l(or)g(example,)g(if)h(deg)p Fn(D)f Fm(\025)f Fr(2)p Fn(g)d Fr(+)f(1,)16 b(then)g Fn(D)q(;)8 b(D)k Fm(\000)e Fn(P)23 b Fr(and)16 b Fn(D)11 b Fm(\000)f Fn(P)17 b Fm(\000)10 b Fn(Q)16 b Fr(all)g(ha)o(v)o(e)g(deg) f Fn(>)f Fr(2)p Fn(g)d Fm(\000)f Fr(2,)0 2447 y(so)16 b(that)h(\(recall)f(that)h(in)f(suc)o(h)g(a)g(case)h(w)o(e)f(ha)o(v)o (e)g Fn(c)p Fr(\()p Fm(\003)p Fr(\))e(=)g(0\))100 2522 y Fn(l)q Fr(\()p Fm(\003)p Fr(\))g(=)g(deg\()p Fm(\003)p Fr(\))e(+)e(1)h Fm(\000)g Fn(g)r Fr(,)100 2598 y(so)j(the)i(ab)q(o)o(v) o(e)f(condition)f(is)h(satis\014ed)f(\(b)q(ecause)h(deg\()p Fn(D)10 b Fm(\000)e Fn(P)16 b Fm(\000)8 b Fn(Q)p Fr(\))16 b(=)f(deg)p Fn(D)h Fr(-)f(deg)p Fn(P)22 b Fr(-)15 b(deg)p Fn(Q)g Fr(=)0 2670 y(deg)p Fn(D)j Fr(-)e(2,)h(since)p 345 2629 V 16 w Fn(k)e Fr(=)e Fn(k)19 b Fr(implies)c(deg)p Fn(P)23 b Fr(=)16 b(deg)p Fn(Q)h Fr(=)f(1\).)950 2790 y(52)p eop %%Page: 53 53 53 52 bop 100 50 a Fr(So)19 b(w)o(e)g(see)h(that)g(if)f(w)o(e)h(tak)o (e)g(a)f(line)g(bundle)g(of)h(large)e(enough)h(degree)g(\(i.e.)h Fm(L)1660 37 y(\030)1660 52 y Fr(=)1718 50 y Fm(L)p Fr(\()p Fn(D)q Fr(\))h(with)0 120 y(deg)p Fn(D)d Fr(large)e(enough\),)g(then)g (w)o(e)g(will)h(ha)o(v)o(e)e(a)i(closed)f(em)o(b)q(edding)f(in)o(to)h (pro)s(jectiv)o(e)g(space.)100 230 y(W)l(e)g(\014nish)g(this)g (discussion)e(with)i(t)o(w)o(o)h(facts)f(whic)o(h)g(w)o(e)g(will)g(not) h(pro)o(v)o(e:)100 301 y(Recall)g(that)h(the)f(canonical)g(divisor)f Fn(W)876 308 y Fk(\025)920 301 y Fr(is)h(de\014ned,)f(up)h(to)h (divisor)e(class,)h(indep)q(enden)o(tly)f(of)0 370 y(the)h(W)l(eil)f (di\013eren)o(tial)f Fn(\025)f Fm(6)p Fr(=)g(0.)i(W)l(e)h(ha)o(v)o(e)f (that)100 441 y Fm(\017)g Fr([)p Fm(8)p Fn(P)q(;)8 b(l)q Fr(\()p Fn(W)321 448 y Fk(\025)358 441 y Fm(\000)i Fn(P)d Fr(\))15 b Fn(<)e(l)q Fr(\()p Fn(W)614 448 y Fk(\025)640 441 y Fr(\)])k(if)g(and)f(only)g(if)h Fn(g)e(>)f Fr(0)100 512 y Fm(\017)j Fr([)p Fm(8)p Fn(P)q(;)8 b(Q)17 b Fr(\(closed\),)g Fn(l)q Fr(\()p Fn(W)578 519 y Fk(\025)616 512 y Fm(\000)11 b Fn(P)19 b Fm(\000)11 b Fn(Q)p Fr(\))16 b(=)f Fn(l)q Fr(\()p Fn(W)977 519 y Fk(\025)1003 512 y Fr(\))d Fm(\000)f Fr(2])18 b(if)f(and)g(only)g(if)g Fn(C)k Fr(is)16 b(not)i(a)f(h)o(yp)q (erelliptic)0 581 y(curv)o(e.)100 692 y(No)o(w)f(w)o(e)g(come)h(bac)o (k)f(to)h(the)g(theory)l(.)f(W)l(e)h(will)f(talk)h(ab)q(out)f (di\013eren)o(tials:)100 763 y Fp(K\177)-28 b(ahler)18 b(di\013eren)n(tials)k(and)d(W)-5 b(eil)21 b(di\013eren)n(tial)q(s)100 873 y Fr(Ev)o(en)o(tually)l(,)e(w)o(e)h(will)f(get)i(a)f(connection)g (b)q(et)o(w)o(een)f(these)i(t)o(w)o(o)e(t)o(yp)q(es)i(of)f(di\013eren)o (tials)e(\(recall)0 943 y(that)e(w)o(e)g(ha)o(v)o(e)g(already)f(met)h (the)g(W)l(eil)g(di\013eren)o(tials\).)f(Often)h(times)g(w)o(e)f(will)h (obtain)f(isomorphism.)100 1043 y(Let)h Fn(A)h Fr(b)q(e)f(a)h(ring)e
(\(that)i(is,)e(a)h(comm)o(utativ)o(e)f(ring)h(with)g(1\),)g(let)h Fn(B)h Fr(b)q(e)e(an)g Fn(A)p Fr(-Algebra)g(\(that)h(is,)0 1113 y(a)g(ring)e(with)i(a)f(ring)g(homomorphis)o(m)d Fn(\036)h Fr(:)g Fn(A)g Fm(\000)-9 b(!)14 b Fn(B)r Fr(\),)j(and)f(let)h Fn(M)22 b Fr(b)q(e)17 b(a)f Fn(B)r Fr(-mo)q(dule.)100 1184 y(Note)h(that)g Fn(M)22 b Fr(has)16 b(a)h(natural)e Fn(A)p Fr(-mo)q(dule)h(structure,)f(and)h(so)g(do)q(es)h Fn(B)r Fr(.)100 1254 y Fp(De\014nition:)h Fr(an)e Fn(A)p Fp(-deriv)m(ation)i Fn(B)e Fm(\000)-9 b(!)14 b Fn(M)23 b Fr(is)16 b(an)g Fn(A)p Fr(-linear)f(map)h Fn(D)i Fr(suc)o(h)e(that)g Fm(8)p Fn(x;)8 b(y)16 b Fm(2)e Fn(B)r Fr(,)100 1325 y Fn(D)q Fr(\()p Fn(xy)r Fr(\))i(=)d Fn(xD)q Fr(\()p Fn(y)r Fr(\))g(+)e Fn(y)r(D)q Fr(\()p Fn(x)p Fr(\).)100 1425 y(Note:)24 b(in)g(general,)f Fn(D)j Fr(will)e(not)g(b)q(e)h Fn(B)r Fr(-linear.)e(Also,)g(note)i(that)f Fn(D)i Fr(is)e(an)g(additiv) o(e)f(group)0 1495 y(homomorphism)13 b(\(b)o(y)j(de\014nition)g(of)h (\\)p Fn(A)p Fr(-linear)e(map"\),)h(and)g(that)h(the)f(comp)q(osite)g (map)100 1583 y Fn(A)165 1553 y Fk(\036)137 1583 y Fm(\000)-9 b(!)p Fn(B)281 1555 y Fk(D)257 1583 y Fm(\000)h(!)p Fn(M)19 b Fr(=)14 b(0.)100 1654 y(T)l(o)20 b(see)h(this,)f(w)o(e)h(observ)o(e)e (that)i(for)g Fn(x)g Fm(2)g Fn(A)g Fr(and)f Fn(y)j Fm(2)e Fn(B)r Fr(,)g(w)o(e)g(m)o(ust)e(ha)o(v)o(e)i(\(b)o(y)f(de\014nition)g (of)0 1724 y(\\)p Fn(A)p Fr(-linear",)15 b(again\))h Fn(D)q Fr(\()p Fn(\036)p Fr(\()p Fn(x)p Fr(\))p Fn(y)r Fr(\))h(=)c Fn(\036)p Fr(\()p Fn(x)p Fr(\))p Fn(D)q Fr(\()p Fn(y)r Fr(\).)100 1794 y(T)l(aking)j Fn(y)g Fr(=)d(1)k(w)o(e)f(ha)o(v)o (e)g Fn(D)q Fr(\()p Fn(\036)p Fr(\()p Fn(x)p Fr(\)\))g(=)e Fn(\036)p Fr(\()p Fn(x)p Fr(\))p Fn(D)q Fr(\(1\))i(=)d(0,)k(as)f (claimed,)f(b)q(ecause)100 1865 y Fn(D)q Fr(\(1\))g(=)e Fn(D)q Fr(\(1)f Fm(\001)f Fr(1\))j(=)g Fn(D)q Fr(\(1\))e(+)f Fn(D)q Fr(\(1\))18 b(implies)d(that)i Fn(D)q Fr(\(1\))e(=)f(0.)100 1965 y(Note)19 b(that)h(if)f Fn(D)g Fr(:)f Fn(B)i Fm(\000)-9 b(!)18 b Fn(M)24 b Fr(is)19 b(an)f Fn(A)p Fr(-deriv)m(ation)h(and)f Fn(\034)23 b Fr(:)18 b Fn(M)23 b Fm(\000)-8 b(!)17 b Fn(M)1514 1947 y Fj(0)1548 1965 y Fr(is)h(a)h Fn(B)r Fr(-linear)f(map,)0 2035 y(then)e(the)h(comp)q(osite)f(map)100 2106 y Fn(\034)g Fm(\016)11 b Fn(D)16 b Fr(:)d Fn(B)j Fm(\000)-8 b(!)13 b Fn(M)459 2088 y Fj(0)491 2106 y Fr(is)j(an)g Fn(A)p Fr(-deriv)m(ation)g(from)g Fn(B)j Fr(to)e Fn(M)1187 2088 y Fj(0)1201 2106 y Fr(.)100 2216 y Fp(Univ)n(ersal)j(Deriv)m (ation:)100 2317 y Fr(De\014ne)c(the)h Fn(B)r Fr(-mo)q(dule)e(\012)608 2326 y Fk(B)r(=)n(A)707 2317 y Fr(b)o(y)i(generators)e(and)h(relations) f(as)h(follo)o(ws:)100 2387 y(generators:)f(one)h(sym)o(b)q(ol,)f Fn(db)p Fr(,)i(for)f(eac)o(h)g Fn(b)f Fm(2)f Fn(B)r Fr(.)100 2458 y(relations:)h Fn(d)p Fr(\()p Fn(b)d Fr(+)e Fn(b)464 2440 y Fj(0)479 2458 y Fr(\))k(=)g Fn(db)d Fr(+)g Fn(db)720 2440 y Fj(0)751 2458 y Fr(for)16 b(all)g Fn(b;)8 b(b)960 2440 y Fj(0)989 2458 y Fm(2)14 b Fn(B)r Fr(.)315 2529 y Fn(da)h Fr(=)e(0)k(for)f(all)g Fn(a)e Fm(2)g Fn(\036)p Fr(\()p Fn(A)p Fr(\).)315 2599 y Fn(d)p Fr(\()p Fn(bb)402 2581 y Fj(0)418 2599 y Fr(\))g(=)f Fn(b)f Fm(\001)f Fn(db)608 2581 y Fj(0)633 2599 y Fr(+)g Fn(b)704 2581 y Fj(0)729 2599 y Fm(\001)g Fn(db)17 b Fr(for)f(all)g Fn(b;)8 b(b)1027 2581 y Fj(0)1056 2599 y Fm(2)14 b Fn(B)r Fr(.)100 2670 y Fp(De\014nition:)k Fr(\012)417 2679 y Fk(B)r(=)n(A)516 2670 y Fr(is)e(the)h(mo)q(dule)e(of)i Fp(K\177)-28 b(ahler)18 b(di\013eren)n(tial)q(s)h Fr(of)e Fn(B)i Fr(o)o(v)o(er)c Fn(A)p Fr(.)950 2790 y(53)p eop %%Page: 54 54
54 53 bop 100 50 a Fr(Note)14 b(that)g(the)f(\(ob)o(vious\))g(map)g Fn(d)g Fr(:)h Fn(B)i Fm(\000)-8 b(!)13 b Fr(\012)979 59 y Fk(B)r(=)n(A)1075 50 y Fr(is)g(a)h(deriv)m(ation)e(\(simply)h(b)q (ecause)g(w)o(e)g(forced)0 117 y(it)k(to)g(b)q(e)f(the)h(case\).)100 187 y(It)g(is)f(a)g(univ)o(ersal)f(deriv)m(ation,)h(in)g(the)h(sense)f (that)g(for)h(all)f Fn(B)r Fr(-mo)q(dules)f Fn(M)5 b Fr(,)17 b(the)g(map)100 257 y(Hom)204 264 y Fk(B)238 257 y Fr(\(\012)293 266 y Fk(B)r(=)n(A)376 257 y Fn(;)8 b(M)d Fr(\))15 b Fm(\000)-9 b(!)14 b Fr(Der)658 264 y Fk(A)691 257 y Fr(\()p Fn(B)r(;)8 b(M)d Fr(\),)18 b(giv)o(en)e(b)o(y)h Fn(g)e Fm(7\000)-9 b(!)14 b Fn(g)f Fm(\016)e Fn(d)p Fr(,)16 b(is)g(bijectiv)o(e.)100 324 y(Here)g(Der)297 331 y Fk(A)330 324 y Fr(\()p Fn(B)r(;)8 b(M)d Fr(\))18 b(is)e(the)h(set)g(of)f Fn(A)p Fr(-deriv)m(ations)g Fn(B)g Fm(\000)-8 b(!)13 b Fn(M)5 b Fr(.)100 394 y(So,)23 b(if)g(placed)h(in)f(the)h(righ)o(t)e (category)l(.)i(\012)944 402 y Fk(B)r(=)n(A)1050 394 y Fr(is)f(a)h(univ)o(ersal)d(ob)s(ject,)j(hence)g(it)g(is)f(unique)0 461 y(\(satisfying)16 b(the)h(univ)o(ersal)e(prop)q(ert)o(y\))h(up)g (to)h(isomp)q(ophism)d(.)100 560 y(Let)j Fn(k)e Fm(\032)f Fn(K)20 b Fr(b)q(e)d(a)g(\014eld)f(extension.)g(Then)g(one)g(can)g(c)o (hec)o(k)h(that)100 630 y(Der)179 637 y Fk(k)204 630 y Fr(\()p Fn(K)q(;)8 b(K)t Fr(\))368 617 y Fm(\030)368 633 y Fr(=)423 630 y(\()p Fn(K)t Fr(-dual)15 b(of)i(\012)708 639 y Fk(K=k)787 630 y Fr(\).)100 737 y(Supp)q(ose)g(that)i Fn(B)i Fr(is)e(giv)o(en)f(b)o(y)g(a)h(\\presen)o(tation")e Fn(B)j Fr(=)d Fn(A)p Fr([)p Fn(X)1308 744 y Fk(i)1325 737 y Fm(j)p Fn(i)g Fm(2)h Fn(I)t Fr(])p Fn(=)p Fr(\()p Fn(f)1532 744 y Fk(j)1553 737 y Fm(j)p Fn(j)i Fm(2)e Fn(J)5 b Fr(\).)18 b(Then)g(one)0 807 y(can)e(sho)o(w)g(without)g(m)o (uc)o(h)f(di\016cult)o(y)h(\(although)f(w)o(e)i(will)f(not)g(do)h(so)f (here\))g(that)414 961 y(\012)450 970 y Fk(B)r(=)n(A)547 961 y Fr(=)599 891 y Fh(\022)636 914 y(X)665 1020 y Fk(i)716 961 y Fn(B)r Fr([)p Fn(dX)837 968 y Fk(i)854 961 y Fr(])868 891 y Fh(\023\036\022)993 961 y Fn(B)1042 914 y Fh(X)1071 1020 y Fk(i)1134 927 y Fn(@)s(f)1187 934 y Fk(j)p 1128 950 88 2 v 1128 995 a Fn(@)s(X)1198 1002 y Fk(i)1221 961 y Fr([)p Fn(dX)1302 968 y Fk(i)1319 961 y Fr(])1349 889 y Fh(\014)1349 919 y(\014)1349 949 y(\014)1349 978 y(\014)1382 961 y Fn(j)h Fm(2)d Fn(J)1500 891 y Fh(\023)100 1095 y Fr(Here)i Fn(dX)284 1102 y Fk(i)318 1095 y Fr(is)g(the)h(formal) e(sym)o(b)q(ol)h(asso)q(ciated)g(to)h Fn(X)1120 1102 y Fk(i)1137 1095 y Fr(.)100 1165 y(Since)h(\012)268 1174 y Fk(B)r(=)n(A)369 1165 y Fr(is)h(a)g(univ)o(ersal)e(ob)s(ject,)i(w)o (e)f(see)h(that)h(the)f(righ)o(t-hand)d(side)i(of)h(this)g(equation)g (is)0 1235 y(actually)e(indep)q(enden)o(t)e(of)i(the)f(presen)o(tation) f(that)i(w)o(e)g(ga)o(v)o(e)f Fn(B)r Fr(.)0 1305 y Fp(Lecture)k(26.)c Fr(\(Octob)q(er)h(27,)f(1995\).)0 1374 y(Scrib)q(e:)g(Dylan)g(Th)o (urston)0 1493 y Fp(K\177)-28 b(ahler)18 b(di\013eren)n(tial)q(s)k(and) c(W)-5 b(eil)22 b(di\013eren)n(tials)0 1575 y Fr(This)16 b(section)g(is)g(a)h(surv)o(ey)e(of)i(results.)e(Pro)q(ofs)i(are)f (omitted)g(since)g(they)h(are)50 1644 y Fm(\017)25 b Fr(not)16 b(need)g(in)h(the)f(sequel)50 1714 y Fm(\017)25 b Fr(completely)16 b(elemen)o(tary)-47 1784 y(and)g Fm(\017)25 b Fr(to)q(o)17 b(long.)100 1854 y(F)l(or)c Fn(A)h Fm(!)g Fn(B)j Fr(an)o(y)d(map)f(of)i(rings,)e(last)i(time)f(I)h(de\014ned)e(a) i Fn(B)r Fr(-mo)q(dule)e(\012)1458 1863 y Fk(B)r(=)n(A)1555 1854 y Fr(of)i(K\177)-25 b(ahler)13 b(di\013eren-)0 1923 y(tials.)j(This)g(comes)g(with)g(an)g Fn(A)p Fr(-linear)g(map)g Fn(d)d Fr(:)h Fn(B)i Fm(!)e Fr(\012)1090 1932 y Fk(B)r(=)n(A)1189
1923 y Fr(satisfying)i(the)h(deriv)m(ation)f(prop)q(ert)o(y:)722 2025 y Fn(d)p Fr(\()p Fn(b)788 2032 y Fl(1)811 2025 y Fn(b)832 2032 y Fl(2)855 2025 y Fr(\))e(=)f Fn(b)961 2032 y Fl(1)984 2025 y Fn(db)1031 2032 y Fl(2)1065 2025 y Fr(+)e Fn(b)1136 2032 y Fl(2)1158 2025 y Fn(db)1205 2032 y Fl(1)0 2130 y Fr(and)k(a)h(univ)o(ersal)e(prop)q(ert)o(y:)h(giv) o(en)g Fn(D)g Fr(:)f Fn(B)i Fm(!)e Fn(M)21 b Fr(another)15 b(deriv)m(ation,)g(there)h(is)f(a)h(unique)f Fn(B)r Fr(-linear)0 2199 y(map)h Fn(f)j Fr(:)14 b(\012)218 2208 y Fk(B)r(=)n(A)314 2199 y Fm(!)g Fn(M)22 b Fr(so)16 b(that)h Fn(d)11 b Fm(\016)g Fn(f)20 b Fr(=)14 b Fn(D)q Fr(.)0 2294 y Fq(R)m(efer)m(enc)m(es:)k Fr(E.)d(Kunz,)f Fq(K\177)-25 b(ahler)18 b(Di\013er)m(entials)p Fr(,)d(View)o(eg,)g(1986.)g(Hartshorne,)f Fq(A)o(lgebr)m(aic)j(Ge)m (ometry)p Fr(,)0 2363 y(Chapter)f(I)q(I,)h(Section)f(8,)g(page)h(173.)0 2457 y Fq(Example:)j Fn(B)i Fr(=)e Fn(a)p Fr([)p Fn(X)419 2464 y Fk(i)455 2457 y Fm(j)f Fn(i)h Fm(2)g Fn(I)t Fr(].)f(Then)h(\012) 821 2466 y Fk(B)r(=)n(A)923 2457 y Fr(is)g Fn(B)r Fr(-free)g(on)f Fn(dX)1272 2464 y Fk(i)1289 2457 y Fr(,)h Fn(i)g Fm(2)f Fn(I)24 b Fr(and)19 b(the)i(map)e Fn(d)g Fr(:)g Fn(B)j Fm(!)0 2525 y Fr(\012)36 2534 y Fk(B)r(=)n(A)132 2525 y Fr(=)185 2487 y Fh(L)240 2540 y Fk(i)p Fj(2)p Fk(I)312 2525 y Fn(B)11 b(dX)428 2532 y Fk(i)461 2525 y Fr(is)769 2623 y Fn(d)-8 b(f)19 b Fr(=)882 2575 y Fh(X)888 2682 y Fk(i)p Fj(2)p Fk(I)983 2589 y Fn(@)s(f)p 969 2611 V 969 2657 a(@)s(X)1039 2664 y Fk(i)1062 2623 y Fn(;)8 b(dX)1151 2630 y Fk(i)1168 2623 y Fn(:)950 2790 y Fr(54)p eop %%Page: 55 55 55 54 bop 0 50 a Fr(It's)16 b(an)h(easy)f(c)o(hec)o(k)g(that)h(this)g (is)f(a)g(deriv)m(ation)g(and)g(univ)o(ersal.)0 149 y Fq(R)o(ules:)30 218 y Fr(1\))26 b(F)l(or)15 b Fn(S)h Fm(\032)e Fn(B)19 b Fr(a)d(m)o(ultiplicativ)o(e)f(set,)566 344 y(\012)602 353 y Fk(S)628 343 y Fi(\000)p Fd(1)675 353 y Fk(B)r(=)n(A)771 344 y Fr(=)f Fn(S)858 323 y Fj(\000)p Fl(1)911 344 y Fr(\012)947 353 y Fk(B)r(=)n(A)1051 344 y Fr(\(=)g(\012)1159 353 y Fk(B)r(=)n(A)1253 344 y Fm(\012)1292 351 y Fk(B)1337 344 y Fn(S)1371 323 y Fj(\000)p Fl(1)1424 344 y Fn(B)r Fr(\))100 470 y(In)20 b(particular,)f(if)h Fn(k)j Fr(is)d(a)g(\014eld,)g(then)h(\012)891 479 y Fk(k)q Fl(\()p Fk(X)962 484 y Fc(i)977 479 y Fj(j)p Fk(i)p Fj(2)p Fk(I)s Fl(\))p Fk(=k)1131 470 y Fr(is)f(a)h(v)o(ector)f(space)g(of)h Fn(k)r Fr(\()p Fn(X)1670 477 y Fk(i)1687 470 y Fr(\))g(with)f(basis)100 539 y Fn(dX)167 546 y Fk(i)183 539 y Fr(.)d(In)f(this)g(case,)h(dim)7 b(\012)14 b(=)i(transcendence)f(degree)i(of)f(extension.)30 609 y(2\))26 b(Base)16 b(c)o(hange:)g(giv)o(en)g(a)g(map)g Fn(A)e Fm(!)g Fn(A)831 591 y Fj(0)845 609 y Fr(,)j(there)f(is)g(a)h (natural)e(map)h Fn(A)1417 591 y Fj(0)1445 609 y Fm(!)e Fn(B)1549 591 y Fj(0)1577 609 y Fr(=)f Fn(B)h Fm(\012)1720 616 y Fk(A)1763 609 y Fn(A)1800 591 y Fj(0)1831 609 y Fr(and)769 735 y(\012)p Fn(B)845 714 y Fj(0)859 735 y Fn(=)m(A)918 714 y Fj(0)946 735 y Fr(=)g(\012)1035 744 y Fk(B)r(=)n(A)1129 735 y Fm(\012)1168 742 y Fk(B)1213 735 y Fn(B)1253 714 y Fj(0)1267 735 y Fn(:)30 860 y Fr(3\))26 b(Quotien)o(ts:)14 b(for)h(an)o(y)h(ideal)f Fn(I)i Fm(\032)d Fn(B)r Fr(,)i(set)f Fn(C)i Fr(=)d Fn(B)r(=I)t Fr(.)h(Then)h(\012)1258 869 y Fk(C)q(=)n(A)1354 860 y Fr(is)f(a)h Fn(B)r Fr(-mo)q(dule)e(and)h (the)h(map)100 930 y(from)f(\012)253 939 y Fk(B)r(=)n(A)352 930 y Fr(to)i(\012)449 939 y Fk(C)q(=)n(A)547 930 y Fr(is)f(surjectiv)o (e.)g(Then)g(formally)l(,)724 1055 y(\012)760 1062 y
Fk(C)789 1067 y Fc(A)833 1055 y Fr(=)e(\(\012)941 1064 y Fk(B)r(=)n(A)1035 1055 y Fm(\012)1074 1062 y Fk(B)1119 1055 y Fn(C)t Fr(\))886 1094 y Fh(|)p 908 1094 101 6 v 101 w({z)p 1053 1094 V 101 w(})911 1143 y Fl(\012)939 1150 y Fc(B)q(=)n(A)1011 1143 y Fk(=I)s Fl(\012)1080 1150 y Fc(B)q(=)n(A)1185 1055 y Fn(=)p Fr(\\)p Fn(dI)t Fr(")p Fn(:)100 1266 y Fr(The)i(notation)g(\\)p Fn(dI)t Fr(")g(means)g(that)h(there)f(is)g(an)h(exact)g(sequence)f(of)h Fn(C)t Fr(-mo)q(dules)668 1403 y Fn(I)t(=I)745 1382 y Fl(2)668 1436 y Fh(|)p 690 1436 5 6 v 5 w({z)p 739 1436 V 5 w(})661 1483 y Fk(I)s Fj(\012)713 1488 y Fc(B)742 1483 y Fk(C)796 1375 y(d)781 1403 y Fm(!)8 b Fr(\012)875 1412 y Fk(B)r(=)n(A)969 1403 y Fm(\012)1008 1410 y Fk(B)1053 1403 y Fn(C)17 b Fm(!)d Fr(\012)1206 1412 y Fk(C)q(=)n(A)1300 1403 y Fm(!)g Fr(0)100 1604 y(\(Note)22 b(that)g Fn(d)g Fr(b)q(ecomes)f Fn(C)t Fr(-linear)e(when)i(y)o(ou)g(mo)q(d)g(out)h(b)o (y)f Fn(I)1352 1586 y Fl(2)1374 1604 y Fr(.\))h(F)l(or)e(instance,)h (tak)o(e)h Fn(B)i Fr(=)100 1674 y Fn(A)p Fr([)p Fn(X)192 1681 y Fk(i)223 1674 y Fm(j)13 b Fn(i)h Fm(2)g Fn(I)t Fr(])i(and)g Fn(C)h Fr(=)d Fn(B)r(=)p Fr(\()p Fn(f)695 1681 y Fk(j)731 1674 y Fm(j)f Fn(j)k Fm(2)d Fn(J)5 b Fr(\).)16 b(Then)499 1831 y(\012)535 1840 y Fk(C)q(=)n(A)629 1831 y Fr(=)682 1760 y Fh(\022)718 1783 y(M)725 1890 y Fk(i)p Fj(2)p Fk(I)802 1831 y Fn(C)e Fm(\001)d Fn(dX)944 1838 y Fk(i)961 1760 y Fh(\023\036\022)1086 1783 y(X)1087 1890 y Fk(j)r Fj(2)p Fk(J)1166 1831 y Fn(C)k Fm(\001)1242 1783 y Fh(X)1247 1890 y Fk(i)p Fj(2)p Fk(I)1328 1797 y Fn(@)s(f)1381 1804 y Fk(j)p 1328 1819 75 2 v 1336 1865 a Fn(X)1377 1872 y Fk(i)1417 1831 y Fn(dX)1484 1838 y Fk(i)1500 1760 y Fh(\023)1537 1831 y Fn(:)100 2009 y Fr(\(This)22 b(will)g(b)q(e)h(used)f(to)h(de\014ne)f Fq(smo)m(othness)p Fr(,)j(whic)o(h)c(is)i(a)f(b)q(etter)h(notion)g (than)f Fq(r)m(e)m(gular)h Fr(for)100 2079 y(non-algebraically)14 b(closed)i(base)g(\014elds.\))100 2149 y(Hartshorne)f(also)h(considers) f(the)i(situation)f(of)h(three)g(rings)e(with)i(maps)f Fn(A)e Fm(!)g Fn(B)j Fm(!)d Fn(C)t Fr(.)i(There)0 2219 y(are)21 b(three)h(K\177)-25 b(ahler)21 b(di\013eren)o(tials:)f(\012) 714 2228 y Fk(B)r(=)n(A)796 2219 y Fr(,)i(\012)868 2228 y Fk(C)q(=B)952 2219 y Fr(,)g(and)f(\012)1126 2228 y Fk(C)q(=)n(A)1207 2219 y Fr(,)g(with)h(an)f(exact)i(sequence)f (relating)0 2288 y(them:)612 2358 y(\012)648 2367 y Fk(B)r(=)n(A)741 2358 y Fm(\012)11 b Fn(C)17 b Fm(!)d Fr(\012)944 2367 y Fk(C)q(=)n(A)1038 2358 y Fm(!)g Fr(\012)1138 2367 y Fk(C)q(=B)1236 2358 y Fm(!)g Fr(0)p Fn(:)0 2461 y Fr(\(The)j(map)e(b)q (et)o(w)o(een)i(the)f(last)h(t)o(w)o(o)f Fn(C)t Fr(-mo)q(dules)e(is)i (de\014ned)g(b)o(y)g(the)g(univ)o(ersal)f(prop)q(ert)o(y)l(,)h(since)g (an)o(y)0 2531 y Fn(B)r Fr(-linear)f(deriv)m(ation)h(is)g(also)g Fn(A)p Fr(-linear.\))100 2600 y(The)h(ultra-classical)d(case)j(of)h (this)e(theory)h(is)g Fp(K\177)-28 b(ahler)19 b(di\013eren)n(tials)j (of)d(\014eld)i(extensions)p Fr(.)0 2670 y(Supp)q(ose)15 b Fn(k)h Fm(\032)d Fn(K)21 b Fr(is)16 b(an)o(y)g(\014eld)g(extension;)g (then)g(the)h(problem)e(is)h(to)h(in)o(v)o(estigate)f(\012)1626 2679 y Fk(K=k)1705 2670 y Fr(.)950 2790 y(55)p eop %%Page: 56 56 56 55 bop 0 50 a Fq(Question:)17 b Fr(What)g(is)f(the)h(dimension)e(of) h(\012)829 59 y Fk(K=k)925 50 y Fr(\(as)g(a)h Fn(K)t Fr(-v)o(ector)f(space\)?)0 143 y(c)o(har)7 b Fn(k)15 b Fr(=)f(0)p 0 151 220 2 v 100 212 a(By)h(general)f(\014eld)g(theory)l (,)h Fn(K)j Fr(is)d(an)f(algebraic)g(sep)q(erable)g(extension)g(of)h
Fn(k)1517 219 y Fl(0)1553 212 y Fr(=)f Fn(k)r Fr(\()p Fn(X)1694 219 y Fk(i)1725 212 y Fm(j)f Fn(i)h Fm(2)g Fn(I)t Fr(\))h(for)0 282 y(some)j Fn(I)t Fr(.)g(Since)g(the)h(c)o (haracteristic)f(is)g(0,)g(ev)o(ery)h(generator)e(of)i Fn(K)q(=k)1333 289 y Fl(0)1374 282 y Fr(has)f(a)h(sep)q(erable)e (irrudicible)0 352 y(p)q(olynomial.)d(By)i(adding)e(these)i(generators) e(one)h(b)o(y)g(one,)g(at)h(eac)o(h)f(stage)h(getting)f(a)h(new)f (generator)0 422 y(and)h(relation,)f(one)i(can)f(sho)o(w)g(that)743 521 y(\012)779 530 y Fk(K=k)872 521 y Fr(=)d(\012)960 530 y Fk(k)981 535 y Fd(0)1001 530 y Fk(=k)1057 521 y Fm(\012)1096 528 y Fk(k)1117 533 y Fd(0)1150 521 y Fn(K)q(:)100 616 y Fr(The)k(follo)o(wing)g(three)g(facts)h(are)f(immediate:)g(\(1\)) h(If)g Fn(x)1165 623 y Fl(1)1188 616 y Fn(;)8 b(:)g(:)g(:)g(;)g(x)1326 623 y Fk(n)1370 616 y Fm(2)16 b Fn(K)21 b Fr(then)d Fn(dx)1651 623 y Fl(1)1674 616 y Fn(;)8 b(:)g(:)g(:)g(;)g(dx)1838 623 y Fk(n)1883 616 y Fr(are)0 686 y(linearly)21 b(indep)q(enden)o(t)g (o)o(v)o(er)g Fn(K)26 b Fr(if)c(and)g(only)f(if)i Fn(x)1000 693 y Fl(1)1022 686 y Fn(;)8 b(:)g(:)g(:)h(;)f(x)1161 693 y Fk(n)1211 686 y Fr(are)21 b(algebraically)g(indp)q(enden)o(t)g(o) o(v)o(er)0 756 y Fn(K)t Fr(.)f(\(2\))g(dim)246 763 y Fk(K)293 756 y Fr(\012)329 765 y Fk(K=k)427 756 y Fr(=)f(tr)8 b(deg[)p Fn(K)q(=k)r Fr(].)19 b(\(3\))i Fn(dx)e Fr(=)g(0)h(if)g(and)f (only)h(if)g Fn(x)g Fr(is)f(algebraic)g(o)o(v)o(er)g Fn(k)r Fr(.)g(When)0 822 y(c)o(har)7 b Fn(k)19 b Fr(=)e Fn(p)g(>)g Fr(0,)h(facts)h(\(1\))g(and)f(\(3\))h(fail)f(badly;)g(but,)g (o)q(ddly)h(enough,)e(\(2\))i(is)f(still)h(true)f(if)g Fn(K)q(=k)j Fr(is)0 892 y(sep)q(erably)16 b(generated.)0 985 y(c)o(har)7 b Fn(k)15 b Fr(=)f Fn(p)g(>)g Fr(0)p 0 1003 312 2 v 100 1055 a(In)i(this)g(case,)g(for)h(an)o(y)f Fn(\013)e Fm(2)g Fn(K)20 b Fr(w)o(e)c(ha)o(v)o(e)757 1154 y Fn(d\013)815 1133 y Fk(p)852 1154 y Fr(=)d Fn(p\013)961 1133 y Fk(p)p Fj(\000)p Fl(1)1044 1154 y Fn(d\013)h Fr(=)f(0)0 1253 y(so)18 b(\012)99 1262 y Fk(K=k)196 1253 y Fr(=)f(\012)288 1262 y Fk(K=k)q Fj(\001)p Fk(K)410 1252 y Fc(p)436 1253 y Fr(;)h(i.e.,)h(w)o(e)f(can)h(reduce)f(immediately)g(to)h(the)g(case)g (of)g(an)g(algebraic,)e(hea)o(vily)0 1322 y(insep)q(erable)e (extension.)h(If)h Fn(x)571 1329 y Fl(1)594 1322 y Fn(;)8 b(:)g(:)g(:)g(;)g(x)732 1329 y Fk(n)774 1322 y Fm(2)14 b Fn(K)t Fr(,)i(then)116 1421 y Fn(dx)170 1428 y Fl(1)193 1421 y Fn(;)8 b(:)g(:)g(:)g(;)g(dx)357 1428 y Fk(n)402 1421 y Fr(linearly)15 b(indep)q(enden)o(t)g(o)o(v)o(er)h Fn(K)i Fm(\()-8 b(\))13 b Fr([)p Fn(k)f Fm(\001)f Fn(K)1257 1401 y Fk(p)1280 1421 y Fr(\()p Fn(x)1327 1428 y Fl(1)1350 1421 y Fn(;)d(:)g(:)g(:)h(;)f(x)1489 1428 y Fk(n)1517 1421 y Fr(\))14 b(:)g Fn(k)e Fm(\001)f Fn(K)1687 1401 y Fk(p)1710 1421 y Fr(])j(=)f Fn(n:)0 1520 y Fr(\(This)25 b(is)g(the)h(basis)e(of)i(the)g(sub)s(ject)f(of)h(\\)p Fn(p)p Fr(-indep)q(endence."\))e(F)l(or)g(instance,)h([)p Fn(K)33 b Fr(:)28 b Fn(k)19 b Fm(\001)e Fn(K)1846 1502 y Fk(p)1869 1520 y Fr(])28 b(=)0 1587 y Fn(p)25 1569 y Fl(dim)91 1574 y Fc(K)130 1569 y Fl(\012)158 1576 y Fc(K=k)230 1587 y Fr(.)16 b(\(When)h(the)g(exp)q(onen)o(t)f(is)g(not)h (\014nite,)f(set)h Fn(p)1108 1569 y Fl(cardinal)1264 1587 y Fr(=)c(cardinal)o(.\))100 1657 y(One)j(can)h(sho)o(w)f(that)h (if)g(c)o(har)c(=)h Fn(p)h(>)f Fr(0)j(and)f Fn(K)q(=k)i Fr(is)f(\014nitely)f(generated,)h(then)f(dim)1735 1664 y Fk(K)1782 1657 y Fr(\012)1818 1666 y Fk(K=k)1911 1657 y Fm(\025)0 1723 y Fr(tr)8 b(deg)h Fn(K)q(=k)r Fr(,)19 b(with)g(equalit)o(y)g(if)h(and)f(only)g(if)g Fn(K)q(=k)i
Fr(is)e(sep)q(erably)g(generated;)f(in)h(particular,)f(if)i Fn(k)h Fr(is)0 1793 y(p)q(erfect.)15 b(The)f(condition)g(\\\014nitely)g (generated")g(is)f(necessary)l(.)h(F)l(or)f(instance,)h(if)g Fn(K)k Fr(is)c(a)h(p)q(erfect)g(\014eld)0 1863 y(in)h(c)o(har)7 b Fn(k)16 b Fm(6)p Fr(=)d(0,)k(then)f(for)g(all)g Fn(\014)h Fm(2)d Fn(K)t Fr(,)i Fn(d\014)h Fr(=)d(0.)0 1932 y Fp(Pro)r(of)k Fr(that)i(if)g Fn(k)h Fr(is)e(p)q(erfect)i(and)e Fn(K)q(=k)i Fr(is)e(\014nitely)h(generated,)f([)p Fn(K)k Fr(:)18 b Fn(k)d Fm(\001)e Fn(K)1497 1914 y Fk(p)1520 1932 y Fr(])19 b(=)g Fn(p)1636 1914 y Fl(tr)6 b(deg\()p Fk(K=k)q Fl(\))1844 1932 y Fr(.)19 b(Let)0 2002 y Fn(x)28 2009 y Fl(1)51 2002 y Fn(;)8 b(:)g(:)g(:)g(;)g(x)189 2009 y Fk(n)237 2002 y Fr(b)q(e)21 b(a)f(transcendence)e(basis)h(for)h Fn(K)k Fr(o)o(v)o(er)19 b Fn(k)j Fr(and)d(set)h Fn(k)1315 2009 y Fl(0)1357 2002 y Fr(=)g Fn(k)r Fr(\()p Fn(x)1491 2009 y Fl(1)1514 2002 y Fn(;)8 b(:)g(:)g(:)g(;)g(x)1652 2009 y Fk(n)1680 2002 y Fr(\).)20 b(The)g(\014elds)0 2072 y Fn(K)t Fr(,)c Fn(K)122 2054 y Fk(p)145 2072 y Fr(,)h Fn(k)202 2079 y Fl(0)224 2072 y Fr(,)g(and)f Fn(k)380 2048 y Fk(p)378 2085 y Fl(0)416 2072 y Fr(=)e Fn(k)r Fr(\()p Fn(x)544 2048 y Fk(p)544 2085 y Fl(1)567 2072 y Fn(;)8 b(:)g(:)g(:)h(;)f(x)706 2054 y Fk(p)706 2084 y(n)734 2072 y Fr(\))17 b(form)f(a)g(square)g(of)h(\014eld)f (extensions.)g(The)h Fn(p)p Fr('th)f(p)q(o)o(w)o(er)g(map)0 2142 y(tak)o(es)h(the)f(\014eld)g(extension)h Fn(K)q(=k)636 2149 y Fl(0)674 2142 y Fr(to)g Fn(K)781 2123 y Fk(p)804 2142 y Fn(=k)857 2118 y Fk(p)855 2155 y Fl(0)880 2142 y Fr(,)f(so)g(in)g(particular)f([)p Fn(K)j Fr(:)c Fn(k)1387 2149 y Fl(0)1409 2142 y Fr(])g(=)f([)p Fn(K)1549 2123 y Fk(p)1586 2142 y Fr(:)g Fn(k)1641 2118 y Fk(p)1639 2155 y Fl(0)1664 2142 y Fr(];)j(therefore)485 2268 y([)p Fn(K)i Fr(:)c Fn(K)633 2248 y Fk(p)656 2268 y Fr(])f(=)753 2235 y([)p Fn(K)18 b Fr(:)13 b Fn(k)882 2211 y Fk(p)880 2248 y Fl(0)905 2235 y Fr(])p 742 2257 189 2 v 742 2302 a([)p Fn(K)802 2288 y Fk(p)838 2302 y Fr(:)h Fn(k)894 2278 y Fk(p)892 2316 y Fl(0)917 2302 y Fr(])950 2268 y(=)1009 2235 y([)p Fn(K)j Fr(:)d Fn(k)1136 2242 y Fl(0)1158 2235 y Fr(][)p Fn(k)1212 2242 y Fl(0)1248 2235 y Fr(:)g Fn(k)1304 2211 y Fk(p)1302 2248 y Fl(0)1326 2235 y Fr(])p 1009 2257 332 2 v 1080 2302 a([)p Fn(K)1140 2288 y Fk(p)1177 2302 y Fr(:)f Fn(k)1232 2278 y Fk(p)1230 2316 y Fl(0)1255 2302 y Fr(])1360 2268 y(=)g Fn(p)1437 2248 y Fk(n)0 2391 y Fr(as)j(desired.)100 2461 y(As)22 b(a)g(sp)q(ecial)g(case,)g(for)g (function)g(\014elds)g(o)o(v)o(er)f(p)q(erfect)i Fn(k)r Fr(,)f(the)h(K\177)-25 b(ahler)21 b(di\013eren)o(tials)g(are)g(1-)0 2531 y(dimensional.)h(\(This)i(is)h(also)f(true)g(for)h(sep)q(erably)f (generated)g(function)g(\014elds,)g(but)g(only)h(when)0 2600 y Fn(k)g Fr(is)d(p)q(erfect)i(is)e(there)h(a)g(canonical)f (isomorphism)e(of)j(the)g(K\177)-25 b(ahler)22 b(di\013eren)o(tials)f (with)i(the)g(W)l(eil)0 2670 y(di\013eren)o(tials.\))950 2790 y(56)p eop %%Page: 57 57 57 56 bop 0 50 a Fp(Lecture)20 b(27)c Fr(\(Octob)q(er)h(30,)f(1995\).)0 120 y(Scrib)q(e:)g(Nghi)g(Nguy)o(en)100 244 y Fp(Kahler)j(Di\013eren)n (tial)q(s)i(vs)e(W)-5 b(eil)22 b(Di\016eren)n(tials)100 353 y(Relev)m(an)n(t)e(fact)f(from)e(last)i(time:)e Fr(Supp)q(ose)e (K/k)h(is)g(a)h(\014nitely)f(generated)g(\014eld)f(extension.)0 423 y(Then)h Fn(dim)217 430 y Fl(K)250 423 y Fr(\012)286 432 y Fl(K)p Fk(=)p Fl(k)374 423 y Fm(\025)e Fr(tr)p
Fn(:)p Fr(deg\(K)p Fn(=)p Fr(k\))i(with)h(equalit)o(y)f(o)q(ccurs)g (i\013)h(K/k)f(is)g(separably)g(generated.)100 532 y Fp(An)22 b(example)g(of)g(Kahler)g(di\013eren)n(tial)q(s)j(from)20 b(n)n(um)n(b)r(er)g(theory:)f Fr(Let)h Fm(O)h Fr(b)q(e)e(the)h(ring)0 602 y(of)i(in)o(tegers)f(in)h(an)g(algebraic)f(n)o(um)o(b)q(er)f (\014eld.)h(Then)h(\012)1093 611 y Fj(O)q Fk(=)p Fg(Z)1199 589 y Fm(\030)1199 605 y Fr(=)1238 609 y Fj(O)1296 602 y Fm(O)q Fn(=)p Fm(D)1400 611 y Fj(O)q Fk(=)p Fg(Z)1483 602 y Fr(,)g(here)g Fm(D)i Fr(denotes)e(the)0 672 y(di\013eren)o(t.)100 782 y Fp(De\014nition:)h Fr(a)f(p)q(oin)o(t)g(P)g(on)g(an)g (n-dimensional)d(v)m(ariet)o(y)j(X)h(o)o(v)o(er)e(a)i(\014eld)e(k)i(is) e(called)h(non-)0 851 y(singular)15 b(or)h(smo)q(oth)g(if)g(\012)507 858 y Fj(O)539 865 y Fd(P)p Fc(=)p Fd(k)619 851 y Fr(is)g(free)h(of)f (rank)g(n)h(as)f(an)g Fm(O)1151 858 y Fl(P)1180 851 y Fr(-mo)q(dule.)100 920 y Fp(Remarks:)100 990 y Fr(i\))k(If)g(P)g(is)f (non-singular,)e(then)i Fm(O)772 997 y Fl(P)821 990 y Fr(is)h(a)f(regular)g(ring.)f(And)i(the)g(con)o(v)o(erse)f(is)g(true)h (if)g(k)g(is)f(a)0 1059 y(p)q(erfect)e(\014eld.)100 1129 y(ii\))d(Supp)q(ose)f(K/k)h(is)g(a)g(\014nitely)g(generated)f(\014eld)h (extension)g(of)g(transcendence)f(degree)h(1.)g(Then)0 1199 y(\012)36 1208 y Fl(K)p Fk(=)p Fl(k)131 1185 y Fm(\030)131 1201 y Fr(=)169 1206 y Fl(K)223 1199 y Fr(K)20 b(\(so)h(the)f(generic)g (p)q(oin)o(t)g(is)g(alw)o(a)o(ys)g(non-singular.)o(\).)e(If)j(P)f Fm(2)h Fr(C\(K\))g(is)f(non-singular,)0 1269 y(then)c(\012)149 1276 y Fj(O)181 1283 y Fd(P)p Fc(=)p Fd(k)258 1255 y Fm(\030)258 1271 y Fr(=)297 1276 y Fj(O)329 1281 y Fd(P)370 1269 y Fm(O)410 1276 y Fl(P)439 1269 y Fr(.)100 1377 y(F)l(or)i(the)i(rest)g(of)g(the)g(lecture)g(w)o(e)f(will)h(assume)e (that)i(there)g(exists)g(a)g(t)g Fm(2)g Fr(K)g(suc)o(h)e(that)j(K)e Fm(\033)0 1447 y Fr(k\(t\))e(is)d(\014nite)h(separable.)f(Then)h(dim) 718 1454 y Fl(K)759 1447 y Fr(\012)795 1456 y Fl(K)p Fk(=)p Fl(k)884 1447 y Fr(=)g(1.)g(W)l(e)h(w)o(an)o(t)e(to)i(compare)e (\012)1491 1456 y Fl(K)p Fk(=)p Fl(k)1580 1447 y Fr(and)h(Di\013)1761 1454 y Fl(K)1793 1447 y Fr(.)h(There)0 1516 y(is)i(a)h(natural)f (homomorphis)o(m)e(\012)663 1523 y Fj(O)695 1530 y Fd(P)p Fc(=)p Fd(k)775 1516 y Fm(\000)-8 b(!)17 b Fr(\012)909 1523 y Fj(O)941 1530 y Fd(P)p Fc(=)p Fd(k)1004 1516 y Fm(\012)1043 1523 y Fj(O)1075 1528 y Fd(P)1102 1516 y Fr(K)g(=)g(\012)1250 1525 y Fl(K)p Fk(=)p Fl(k)1342 1516 y Fr(=)g(K)p Fn(:)p Fr(dt.)h(Since)h Fm(O)1703 1523 y Fl(P)1750 1516 y Fr(is)g(a)f(PID,)0 1594 y(\012)36 1601 y Fj(O)68 1608 y Fd(P)p Fc(=)p Fd(k)145 1580 y Fm(\030)145 1596 y Fr(=)198 1594 y Fm(O)238 1601 y Fl(P)266 1572 y Fj(\010)p Fk(:::)344 1594 y Fm(\010)8 b Fr(\(torsion)o(\).)15 b(Therefore)g(the)g(image)g(of)g(\012)1151 1601 y Fj(O)1183 1608 y Fd(P)p Fc(=)p Fd(k)1261 1594 y Fr(in)g(\012)1354 1603 y Fl(K)p Fk(=)p Fl(k)1443 1594 y Fr(is)g(free)g(of)g(rank)g(1.)g (Infact,)0 1665 y(for)i(almost)f(all)h(P)g(one)g(has)f(\012)576 1672 y Fj(O)608 1679 y Fd(P)p Fc(=)p Fd(k)686 1652 y Fm(\030)686 1667 y Fr(=)740 1665 y Fm(O)780 1672 y Fl(P)808 1665 y Fn(:)p Fr(dt)i(\(hence)f(P)g(is)f(smo)q(oth.\))h(So)g(\012)1462 1674 y Fl(K)p Fk(=)p Fl(k)1553 1665 y Fr(plus)f(the)i(images)e(of)0 1733 y(the)h(\012)122 1740 y Fj(O)154 1747 y Fd(P)p Fc(=)p Fd(k)217 1733 y Fr('s)f(form)g(a)g(line)g(bundle.)100 1845 y Fp(Theorem:)k Fr(Supp)q(ose)i(that)h(K/k)f(is)h(separably)e (generated)h(of)h(tr.deg=1.)e(Then)h(there)h(is)f(a)0 1914 y(K-v)o(ector)17 b(space)h(isomorphism)d(c:\012)706 1923 y Fl(K)p Fk(=)p Fl(k)797 1914 y Fm(\000)-9 b(!)16 b Fr(Di\013)978 1921 y Fl(K)1029 1914 y Fr(suc)o(h)h(that)i(c\(\(\012)
1349 1923 y Fl(K)p Fk(=)p Fl(k)1424 1914 y Fr(\))1443 1921 y Fl(p)1468 1914 y Fr(\))e Fm(\022)f Fr(\(Di\013)1663 1921 y Fl(K)1696 1914 y Fr(\))1715 1921 y Fl(p)1758 1914 y Fr(for)i(all)f(P)l(,)0 1984 y(and)f(that)h(the)g(inclusion)d(b)q (ecomes)j(=)f(for)g(all)g(smo)q(oth)g(P)l(.)100 2094 y(Before)i(giving)g(the)h(pro)q(of)f(of)h(this)f(theorem)f(w)o(e)i (recall)e(that)i(Di\013)1385 2101 y Fl(K)1435 2094 y Fr(=)e Fm(f)h Fr(con)o(tin)o(uous)e(k-linear)0 2163 y(maps)h Fp(A)175 2170 y Fl(K)225 2163 y Fm(\000)-9 b(!)17 b Fr(k,)h(zero)g(on)g (K)g Fm(g)h Fr(and)e(\(Di\013)861 2170 y Fl(K)894 2163 y Fr(\))913 2170 y Fl(P)960 2163 y Fr(=)f Fm(f)i Fr(elemen)o(ts)g(in)g (Di\013)1408 2170 y Fl(K)1460 2163 y Fr(whic)o(h)f(are)h(also)f(zero)h (on)22 2224 y(^)0 2237 y Fm(O)40 2244 y Fl(P)69 2237 y Fm(g)p Fr(.)e(W)l(e)h(also)f(ha)o(v)o(e)g(the)h(follo)o(wing)e(fact:) 376 2354 y(P)i(\(closed)f(p)q(oin)o(t\))e Fm(2)g Fr(C)j(is)f(smo)q(oth) d Fm(\()-8 b(\))13 b Fr(\012)1204 2361 y Fj(O)1236 2368 y Fd(P)p Fc(=)p Fd(k)1313 2340 y Fm(\030)1313 2356 y Fr(=)1366 2354 y(\(\012)1421 2363 y Fl(K)p Fk(=)p Fl(k)1495 2354 y Fr(\))1514 2361 y Fj(O)1546 2366 y Fd(P)448 2472 y Fm(\()-8 b(\))13 b(9)p Fn(\013)h Fm(2)g(O)714 2479 y Fl(P)759 2472 y Fr(suc)o(h)i(that)h Fm(O)1020 2479 y Fl(P)1049 2472 y Fn(=)p Fr(m)p 1074 2480 42 2 v 1115 2485 a Fl(P)1158 2472 y Fr(=)d(k\()p Fn(\013)g Fr(mo)q(d)g(m)p 1412 2480 V 1453 2485 a Fl(P)1482 2472 y Fr(\))123 2571 y(and)i(if)k Fn(f)j Fr(=)13 b(irreducible)i(p)q(olynomial)g(of)21 b(\()p Fn(\013)14 b Fr(mo)q(d)g(m)p 1108 2579 V 1149 2583 a Fk(P)1182 2571 y Fr(\))j(o)o(v)o(er)f(k)h(then)f Fn(f)5 b Fr(\()p Fn(\013)p Fr(\))16 b Fm(2)e Fr(m)p 1643 2579 V 1684 2583 a Fl(P)1713 2571 y Fm(\000)p Fr(m)p 1752 2579 V -21 x Fl(2)1794 2583 y Fk(P)760 2670 y Fr(\(ie)p Fn(;)j Fr(m)p 846 2678 V 887 2682 a Fk(P)935 2670 y Fr(=)c Fm(O)1027 2677 y Fl(P)1056 2670 y Fn(:f)5 b Fr(\()p Fn(\013)p Fr(\)\))950 2790 y(57)p eop %%Page: 58 58 58 57 bop 100 50 a Fr(No)o(w)24 b(w)o(e)g(b)q(egin)g(the)g(pro)q(of)g (of)g(the)h(theorem:)e(\(cf.)i(J.T)l(ate,)f(Residues)e(of)j(Di\013eren) o(tials)e(On)0 120 y(Curv)o(es,)15 b(Ann.)i(Scien)o(t.)f(Ec.)g(Nom.)h (Sup.)e(1)i(\(1968\),)g(pp.)f(149-159,)f(or)h(K.Iw)o(asa)o(w)o(a,)f (Algebraic)h(func-)0 189 y(tions,)g(AMS)g(1993,)g(App)q(endix\))100 259 y(First)24 b(w)o(e)h(need)h(some)f(facts)g(ab)q(out)h(v)o(ector)g (space)f(endomorphism)o(s.)d(Let)27 b(V)f(b)q(e)f(a)h(v)o(ector)0 329 y(space)g(o)o(v)o(er)g(a)h(\014eld)f(k.)h(An)g(endomorphism)c Fn(f)5 b Fr(:)15 b(V)32 b Fm(\000)-9 b(!)31 b Fr(V)d(is)16 b(called)26 b Fn(f)5 b(inite)28 b(potent)f Fr(if)k Fm(9)p Fr(n)f Fm(\025)0 399 y Fr(0)24 b(suc)o(h)15 b(that)24 b Fn(dim)363 406 y Fl(k)386 399 y Fn(f)415 381 y Fl(n)441 399 y Fr(\(V\))h(is)e(\014nite.)g(This)g(is)g(equiv)m(alen)o(t)h(to)g (sa)o(ying)f(that)h(there)g(exists)g(a)f(\014nite)0 468 y(subspace)15 b(W)i Fm(\032)f Fr(V)h(suc)o(h)f(that)h Fn(f)5 b Fr(\(W\))15 b Fm(\032)f Fr(W)j(and)f Fn(f)22 b Fr(is)16 b(nilp)q(oten)o(t)h(on)f(V/W.)100 538 y(Supp)q(ose)f(that)i Fn(f)22 b Fr(is)16 b(\014nite)h(p)q(oten)o(t,)f(w)o(e)h(de\014ne)767 669 y(T)l(r\()p Fn(f)5 b Fr(\))15 b(=)30 b(T)l(r)o(\()p Fn(f)20 b Fm(j)14 b Fr(W\))0 799 y(,)i(where)g(W)h(is)f(the)h(subspace) e(ab)q(o)o(v)o(e.)h(Then)g(the)h(follo)o(wings)e(are)h(true:)100 869 y(i\))e(If)h(V)233 851 y Fj(0)261 869 y Fr(is)e(a)i(subspace)d(of)j (V)f(suc)o(h)f(that)i Fn(f)5 b Fr(\(V)961 851 y Fj(0)976 869 y Fr(\))14 b Fm(\032)g Fr(V)h(then)f(T)l(r)1276 876 y Fl(V)1309 869 y Fr(\()p Fn(f)5 b Fr(\))29 b(=)f(T)l(r)1523
883 y Fl(V)1553 865 y Fi(0)1572 869 y Fr(\()p Fn(f)5 b Fr(\))22 b(+)e(T)l(r)1771 878 y Fl(V)q Fk(=)p Fl(V)1851 868 y Fi(0)1868 869 y Fr(\()p Fn(f)5 b Fr(\).)100 939 y(ii\))22 b(If)g Fn(f)5 b Fr(:)14 b(V)24 b Fm(\000)-9 b(!)22 b Fr(Uand)g Fn(g)r Fr(:)13 b(U)23 b Fm(\000)-9 b(!)23 b Fr(V)g(are)e(suc)o(h)g(that)h Fn(f)5 b(g)24 b Fr(is)d(\014nite-p)q(oten)o(t,)h(then)f Fn(g)r(f)27 b Fr(is)22 b(also)0 1009 y(\014nite-p)q(oten)o(t)16 b(and)g(w)o(e)g(ha) o(v)o(e)g(T)l(r)621 1016 y Fl(V)654 1009 y Fr(\()p Fn(g)r(f)5 b Fr(\))31 b(=)g(T)l(r)899 1016 y Fl(U)931 1009 y Fr(\()p Fn(f)5 b(g)r Fr(\).)100 1118 y Fp(Remark:)15 b Fr(If)i Fn(f)5 b Fr(,)18 b Fn(g)r Fr(,)e(and)g Fn(f)h Fr(+)11 b Fn(g)18 b Fr(are)f(all)f(\014nite-p)q(oten)o(t,)h(it)g(is)f(not)h (alw)o(a)o(ys)f(true)g(that)h(T)l(r\()p Fn(f)g Fr(+)11 b Fn(g)r Fr(\))0 1188 y(=)16 b(T)l(r\()p Fn(f)5 b Fr(\))18 b(+)e(T)l(r\()p Fn(g)r Fr(\).)0 1319 y Fp(Lecture)k(28)c Fr(\(No)o(v)o(em)o(b)q(er)g(1,)g(1995\).)0 1389 y(Scrib)q(e:)g(W)l(a)o (yne)g(Whitney)100 1519 y(As)i(usual,)f(w)o(e)h(let)g Fn(K)i Fm(\033)d Fn(k)i Fr(b)q(e)g(a)f(function)g(\014eld)f(of)h(a)g (curv)o(e,)g(that)g(is)g(w)o(e)g(ask)g(that)h Fn(K)h Fm(\033)c Fn(k)k Fr(b)q(e)0 1589 y(\014nitely)c(generated)g(of)g (transcendence)f(degree)h(1.)g(F)l(urthermore,)d(to)q(da)o(y)j(w)o(e)g (imp)q(ose)g(the)g(condition)0 1659 y(that)h Fn(K)h Fm(\033)13 b Fn(k)18 b Fr(is)e(separably)g(generated.)g(W)l(e)g(will)g(discuss)f (the)i(follo)o(wing)e(theorem)h(of)h(T)l(ate:)0 1789 y Fp(Theorem)e Fr(There)g(exists)i(a)f Fn(K)t Fr(-linear)f(isomorphism) e Fn(c)g Fr(:)h(\012)1154 1798 y Fk(K=k)1246 1789 y Fm(!)g Fr(Di\013)1395 1798 y Fk(K=k)1490 1789 y Fr(suc)o(h)h(that)i Fn(c)f Fr(induces)f(a)0 1859 y(map)g(\012)146 1868 y Fk(O)176 1873 y Fc(P)204 1868 y Fk(=k)263 1859 y Fm(!)f Fr(\(Di\013)431 1868 y Fk(K=k)510 1859 y Fr(\))529 1866 y Fk(P)578 1859 y Fr(for)h(eac)o(h)g Fn(P)21 b Fm(2)14 b Fn(C)t Fr(,)g(and)h(this)g(induced)f(map)h(is)g(an)g(isomorphism)d (if)k(and)0 1929 y(only)g(if)h Fn(P)23 b Fr(is)17 b(a)f(smo)q(oth)g(p)q (oin)o(t.)g(Th)o(us)f(if)i Fn(C)j Fr(is)c(smo)q(oth,)g Fn(c)g Fr(is)g(an)g(isomorphism)d(of)k(line)f(bundles.)0 2060 y Fp(Classical)22 b(Approac)n(h)c(to)h(the)h(Theorem)100 2130 y Fr(Giving)12 b(a)i Fn(K)t Fr(-linear)d(map)h Fn(c)i Fr(:)f(\012)700 2139 y Fk(K=k)793 2130 y Fm(!)h Fr(Di\013)941 2139 y Fk(K=k)1033 2130 y Fr(is)f(equiv)m(alen)o(t)g(to)h(giving)e(a)h Fn(k)r Fr(-linear)f(deriv)m(ation)0 2200 y Fn(D)20 b Fr(:)d Fn(K)22 b Fm(!)c Fr(Di\013)309 2209 y Fk(K=k)387 2200 y Fr(,)h(b)o(y)g(the)g(univ)o(ersal)e(prop)q(ert)o(y)h(of)i(\012) 1095 2209 y Fk(K=k)1173 2200 y Fr(.)f(Then)g(for)f Fn(y)i Fm(2)f Fn(K)t Fr(,)f(w)o(e)h(w)o(an)o(t)f Fn(D)q Fr(\()p Fn(y)r Fr(\))i Fm(2)0 2270 y Fr(Di\013)85 2279 y Fk(K=k)164 2270 y Fr(,)c(that)h(is)f Fn(D)q Fr(\()p Fn(y)r Fr(\))i(should)d(b)q(e) i(a)f(con)o(tin)o(uous)f Fn(k)r Fr(-linear)f(map)i Fp(A)1329 2277 y Fk(K)1382 2270 y Fm(!)d Fn(k)18 b Fr(with)f Fn(K)h Fm(\032)13 b Fr(k)o(er)8 b Fn(D)q Fr(\()p Fn(y)r Fr(\).)100 2340 y(Then)14 b(for)g Fn(x)g Fm(2)g Fp(A)434 2347 y Fk(K)487 2340 y Fr(and)g Fn(y)i Fm(2)e Fn(K)k Fr(as)c(ab)q(o)o(v)o(e)g (w)o(e)h(ha)o(v)o(e)e(a)i(pairing)d(\()p Fn(x;)c(y)r Fr(\))16 b Fm(7!)d Fn(D)q Fr(\()p Fn(y)r Fr(\)\()p Fn(x)p Fr(\);)18 b(this)c(should)0 2410 y(b)q(e)j Fn(k)h Fr(bilinear,)d(and)h (in)g(order)f(to)i(mak)o(e)f Fn(D)i Fr(a)f(deriv)m(ation,)f(it)g (should)f(satisfy)l(,)i(for)f Fn(z)g Fm(2)e Fn(K)t Fr(:)611 2540 y Fn(D)q Fr(\()p Fn(y)r(z)r Fr(\)\()p Fn(x)p Fr(\))j(=)d Fn(D)q Fr(\()p Fn(z)r Fr(\)\()p Fn(xy)r Fr(\))g(+)d Fn(D)q
Fr(\()p Fn(y)r Fr(\)\()p Fn(xz)r Fr(\))0 2670 y Fp(Classical)22 b(Case)d Fn(k)c Fr(=)f Fp(C)950 2790 y Fr(58)p eop %%Page: 59 59 59 58 bop 100 50 a Fr(F)l(or)15 b Fn(x)f Fm(2)g Fp(A)321 57 y Fk(K)360 50 y Fr(,)j(write)f Fn(x)e Fr(=)g(\()p Fn(x)660 57 y Fk(P)694 50 y Fr(\))713 57 y Fk(P)5 b Fj(2)p Fk(C)804 50 y Fr(,)17 b(where)f Fn(x)1007 57 y Fk(P)1054 50 y Fm(2)e Fn(K)1143 57 y Fk(P)1177 50 y Fr(.)i(Then)g(w)o(e)h(use)682 183 y Fn(D)q Fr(\()p Fn(y)r Fr(\)\()p Fn(x)p Fr(\))g(=)932 136 y Fh(X)924 243 y Fk(P)5 b Fj(2)p Fk(C)1021 183 y Fr(res)1082 190 y Fk(P)1115 183 y Fr(\()p Fn(x)1162 190 y Fk(P)1196 183 y Fn(dy)r Fr(\))0 356 y(where)24 b(res)213 363 y Fk(P)246 356 y Fr(\()p Fn(x)293 363 y Fk(P)327 356 y Fn(dy)r Fr(\))h(can)f(b)q(e)g(de\014ned)f(using)g(in)o(tegration) g(or)g(as)h(follo)o(ws:)f Fn(K)1539 363 y Fk(P)1599 343 y Fm(\030)1599 358 y Fr(=)1664 356 y Fn(k)r Fr(\(\()p Fn(t)p Fr(\)\),)i(so)e(w)o(e)0 426 y(can)f(write)g Fn(x)258 433 y Fk(P)316 426 y Fr(=)378 388 y Fh(P)431 441 y Fk(i)456 426 y Fn(a)482 433 y Fk(i)498 426 y Fn(t)516 408 y Fk(i)533 426 y Fr(,)g(and)g(since)g Fn(y)k Fm(2)e Fn(K)j Fm(\032)c Fn(K)1080 433 y Fk(P)1114 426 y Fr(,)f(w)o(e)g(can)h(also)e(write)i Fn(y)i Fr(=)1678 388 y Fh(P)1730 441 y Fk(j)1760 426 y Fn(b)1781 433 y Fk(j)1802 426 y Fn(t)1820 408 y Fk(j)1841 426 y Fr(.)d(Let)0 503 y Fn(dy)16 b Fr(=)119 465 y Fh(P)171 518 y Fk(j)200 503 y Fn(j)s(b)245 510 y Fk(j)266 503 y Fn(t)284 485 y Fk(j)r Fj(\000)p Fl(1)356 503 y Fr(,)h(and)e(w)o(e)i (de\014ne)344 645 y(res)405 652 y Fk(P)438 645 y Fr(\()p Fn(x)485 652 y Fk(P)520 645 y Fn(dy)r Fr(\))d(=)30 b(the)17 b(co)q(e\016cien)o(t)f(of)h Fn(t)1067 627 y Fj(\000)p Fl(1)1137 645 y Fr(in)f Fn(x)1223 652 y Fk(P)1257 645 y Fn(dy)g Fr(=)1396 598 y Fh(X)1375 704 y Fk(i)p Fl(+)p Fk(j)r Fl(=0)1497 645 y Fn(j)s(a)1547 652 y Fk(i)1564 645 y Fn(b)1585 652 y Fk(j)100 822 y Fr(Then)21 b(for)g Fn(D)q Fr(\()p Fn(y)r Fr(\))j(to)e(b)q(e)g(a)f(W)l(eil)h(di\013eren)o (tial,)e(for)h Fn(x)i Fm(2)g Fn(K)i Fr(w)o(e)d(w)o(an)o(t)f Fn(D)q Fr(\()p Fn(y)r Fr(\)\()p Fn(x)p Fr(\))k(=)d(0.)g(This)f(is)0 891 y(tan)o(tamoun)o(t)15 b(to)i(the)g(residue)e(theorem:)h(the)h(sum)e (of)i(the)g(residues)d(of)j(a)g(global)e(di\013eren)o(tial)h(is)g(0.)0 1018 y Fp(Classical)22 b(Case)d Fn(k)c Fm(6)p Fr(=)f Fp(C)100 1088 y Fr(T)l(o)23 b(generalize)g(to)g Fn(k)k Fm(6)p Fr(=)e Fp(C)p Fr(,)f(w)o(e)f(can)g(try)h(to)f(use)g(the)h (algebraic)e(form)o(ula)g(ab)q(o)o(v)o(e)h(to)h(de\014ne)0 1158 y(res)61 1165 y Fk(P)95 1158 y Fr(\()p Fn(x)142 1165 y Fk(P)176 1158 y Fn(dy)r Fr(\).)17 b(Tw)o(o)f(problems)e(ma)o(y)i (arise:)100 1228 y(\(1\))k(The)f(isomorphism)d Fn(K)624 1235 y Fk(P)675 1214 y Fm(\030)675 1230 y Fr(=)732 1228 y Fn(k)r Fr(\(\()p Fn(t)p Fr(\)\))21 b(ma)o(y)d(fail;)h(but)g(it)h (will)e(hold)h(if)g(w)o(e)g(assume)f Fn(k)i Fr(=)1846 1214 y(\026)1845 1228 y Fn(k)q Fr(,)f(so)0 1297 y(w)o(e)d(can)g (\014rst)g(establish)f(the)i(result)e(for)774 1284 y(\026)772 1297 y Fn(k)r Fr(.)h(Then)g(for)g Fn(k)f Fm(6)p Fr(=)1131 1284 y(\026)1130 1297 y Fn(k)j Fr(w)o(e)e(can)g(use)g(Galois)g(theory)g (if)1780 1284 y(\026)1779 1297 y Fn(k)f Fm(\033)f Fn(k)k Fr(is)0 1367 y(Galois)e(\(for)g(example,)g(when)g Fn(k)j Fr(is)d(p)q(erfect\).)100 1437 y(\(2\))i(W)l(e)h(need)e(to)i(sho)o(w)e (that)683 1399 y Fh(P)736 1452 y Fk(i)p Fl(+)p Fk(j)r Fl(=0)860 1437 y Fn(j)s(a)910 1444 y Fk(i)926 1437 y Fn(b)947 1444 y Fk(j)987 1437 y Fr(is)g(indep)q(enden)o(t)g(of)h(the)h (c)o(hoice)e(of)h(isomorphism)0 1510 y Fn(K)42 1517 y
Fk(p)83 1497 y Fm(\030)83 1513 y Fr(=)140 1510 y Fn(k)r Fr(\(\()p Fn(t)p Fr(\)\).)i(It)g(is)f(true)g(if)g(c)o(har)o Fn(k)g Fr(=)f(0)h(b)o(y)g(in)o(tegration.)f(As)h(the)h(desired)d (statemen)o(t)i(is)g(a)g(formal)0 1580 y(iden)o(tit)o(y)d(is)g(some)g (large)g(p)q(olynomial)f(ring)h(o)o(v)o(er)f Fp(Z)p Fr(,)i(then)f(it)h (is)f(true)h(formally)l(.)100 1650 y(\(3\))g(Lastly)l(,)g(w)o(e)f(need) h(to)g(pro)o(v)o(e)e(the)i(residue)f(theorem.)g(W)l(e)h(v)o(erify)g(it) g(for)f Fn(K)i Fr(=)c Fn(k)r Fr(\()p Fn(t)p Fr(\))j(and)f(then)0 1719 y(sho)o(w)f(that)i(it)g(holds)f(for)g(\014nite)g(separable)f (extensions)h(of)h Fn(k)r Fr(\()p Fn(t)p Fr(\).)0 1846 y Fp(T)-5 b(ate's)19 b(Approac)n(h)100 1916 y Fr(Some)f(preliminary)g (linear)g(algebra:)g(Let)j Fn(V)30 b Fr(b)q(e)20 b(a)f(v)o(ector)h (space)f(o)o(v)o(er)g Fn(k)r Fr(;)g(the)g(concepts)h(here)0 1986 y(are)k(in)o(teresting)f(only)i(when)f(dim)683 1993 y Fk(k)716 1986 y Fn(V)38 b Fr(=)27 b Fm(1)p Fr(.)d(Then)g Fn(f)33 b Fm(2)27 b Fr(End)o(\()p Fn(V)12 b Fr(\))25 b(is)f(\\\014nite-p)q(oten)o(t")g(if)g(there)0 2056 y(exists)19 b Fn(n)f Fm(2)h Fp(Z)p Fr(,)g Fn(n)f Fm(\025)g Fr(0)h(suc)o(h)f(that)h (dim)765 2063 y Fk(k)798 2056 y Fn(f)827 2037 y Fk(n)855 2056 y Fn(V)29 b(<)18 b Fm(1)p Fr(.)h(F)l(or)f(a)h(\014nite-p)q(oten)o (t)g(map)f Fn(f)5 b Fr(,)20 b(w)o(e)f(can)g(de\014ne)0 2125 y(T)l(r)8 b Fn(f)27 b Fr(=)21 b(T)l(r)o(\()p Fn(f)5 b Fm(j)284 2132 y Fk(f)307 2123 y Fc(n)333 2132 y Fk(V)367 2125 y Fr(\))22 b(for)e(an)o(y)h Fn(n)g Fr(with)g(dim)839 2132 y Fk(k)872 2125 y Fn(f)901 2107 y Fk(n)928 2125 y Fn(V)33 b(<)21 b Fm(1)p Fr(.)f(This)h(de\014nition)e(is)i(indep)q (enden)o(t)f(of)h(the)0 2195 y(c)o(hoice)h(of)h Fn(n)p Fr(,)g(as)g(for)f Fn(n)464 2202 y Fl(2)511 2195 y Fn(>)i(n)604 2202 y Fl(1)649 2195 y Fr(with)f(dim)852 2202 y Fk(k)884 2195 y Fn(f)913 2177 y Fk(n)938 2182 y Fd(1)961 2195 y Fn(V)35 b(<)24 b Fm(1)p Fr(,)f Fn(f)5 b Fm(j)1218 2202 y Fk(f)1241 2191 y Fc(n)1263 2198 y Fd(1)1286 2202 y Fk(V)1343 2195 y Fr(maps)22 b Fn(f)1509 2177 y Fk(n)1534 2182 y Fd(2)1556 2195 y Fn(V)34 b Fr(to)23 b(itself)g(and)g(is)0 2265 y(nilp)q(oten)o(t)16 b(on)g Fn(f)311 2247 y Fk(n)336 2252 y Fd(1)359 2265 y Fn(V)6 b(=f)448 2247 y Fk(n)473 2252 y Fd(2)494 2265 y Fn(V)12 b Fr(.)0 2392 y Fp(F)-5 b(ormal)18 b(Prop)r(erties)h(of)f(T)-5 b(ate's)19 b Fr(T)l(r)100 2461 y(\(1\))f(If)f Fm(f)p Fr(0)p Fm(g)e(\032)f Fn(W)22 b Fm(\032)15 b Fn(V)28 b Fr(and)16 b Fn(f)5 b Fr(\()p Fn(W)i Fr(\))17 b Fm(\032)d Fn(W)7 b Fr(,)17 b(then)h Fn(f)5 b Fm(j)1085 2468 y Fk(V)1137 2461 y Fr(is)17 b(\014nite-p)q (oten)o(t)31 b Fm(\()-8 b(\))29 b Fn(f)5 b Fm(j)1652 2468 y Fk(W)1715 2461 y Fr(and)17 b Fn(f)5 b Fm(j)1856 2470 y Fk(V)t(=W)0 2531 y Fr(are)16 b(\014nite-p)q(oten)o(t.)g(In)g (this)g(case,)h(T)l(r)712 2538 y Fk(V)755 2531 y Fn(f)j Fr(=)13 b(T)l(r)902 2538 y Fk(W)956 2531 y Fn(f)5 b Fm(j)999 2538 y Fk(W)1056 2531 y Fr(+)11 b(T)l(r)1157 2540 y Fk(V)t(=W)1259 2531 y Fn(f)5 b Fm(j)1302 2540 y Fk(V)t(=W)1396 2531 y Fr(.)100 2600 y(\(2\))24 b(If)f Fn(f)31 b Fr(:)26 b Fn(V)36 b Fm(!)25 b Fn(W)31 b Fr(and)22 b Fn(g)27 b Fr(:)e Fn(W)33 b Fm(!)25 b Fn(V)11 b Fr(,)23 b(then)g Fn(g)r(f)29 b Fr(is)23 b(\014nite-p)q(oten)o(t)g(on)g Fn(V)62 b Fm(\()-8 b(\))50 b Fn(f)5 b(g)26 b Fr(is)0 2670 y(\014nite-p)q(oten)o(t)16 b(on)g Fn(W)7 b Fr(.)17 b(In)f(this)g(case,)g(T)l(r)769 2677 y Fk(V)803 2670 y Fr(\()p Fn(g)r(f)5 b Fr(\))15 b(=)f(T)l(r)1015 2677 y Fk(W)1060 2670 y Fr(\()p Fn(f)5 b(g)r Fr(\).)950 2790 y(59)p eop %%Page: 60 60
60 59 bop 100 50 a Fr(\(3\))16 b(F)l(or)f Fn(f)5 b Fr(,)16 b Fn(g)f Fm(2)g Fr(End)o(\()p Fn(V)c Fr(\),)16 b(it)g(is)g(not)f(true)h (that)g Fn(f)22 b Fr(and)15 b Fn(g)i Fr(\014nite-p)q(oten)o(t)e (implies)g Fn(f)g Fr(+)9 b Fn(g)17 b Fr(is)f(\014nite-)0 120 y(p)q(oten)o(t.)j(So)f(supp)q(ose)f(that)i Fn(f)5 b Fr(,)20 b Fn(g)r Fr(,)e(and)g Fn(f)g Fr(+)12 b Fn(g)21 b Fr(are)d(all)g(\014nite-p)q(oten)o(t.)g(Is)g(it)h(true)g(that)g(T)l (r)o(\()p Fn(f)g Fr(+)12 b Fn(g)r Fr(\))17 b(=)0 189 y(T)l(r)8 b Fn(f)14 b Fr(+)9 b(T)l(r)e Fn(g)r Fr(?)15 b(An)o(y)g(\014nite-p)q(oten)o(t)g(map)g(can)g(b)q(e)h(decomp)q(osed)e (as)h(the)h(sum)e(of)i(a)f(nilp)q(oten)o(t)g(map)g(and)0 259 y(a)i(map)f(with)g(\014nite-dimensional)e(image.)i(Then)h(T)l(ate)g (sho)o(ws)e(it)i(is)f(true)h(if)g(the)g(sub)o(v)o(ector)e(space)i(of)0 329 y(End)o(\()p Fn(V)12 b Fr(\))17 b(spanned)e(b)o(y)h Fn(f)22 b Fr(and)16 b Fn(g)i Fr(can)f(b)q(e)f(sim)o(ultaneously)f (decomp)q(osed.)100 399 y(If)i(there)h(is)e(a)i(coun)o(terexample,)e (then)h(there)g(is)g(a)h(coun)o(terexample)d(with)j Fn(f)23 b Fr(and)17 b Fn(g)i Fr(nilp)q(oten)o(t,)0 468 y(b)o(y)j(the)g(ab)q(o)o (v)o(e)g(decomp)q(osition,)f(so)h(w)o(e)g(need)g(only)g(consider)f (nilp)q(oten)o(t)h(maps.)e(F)l(urthermore,)g(if)0 538 y Fn(f)29 520 y Fl(2)52 538 y Fn(V)25 b Fr(=)14 b Fn(g)185 520 y Fl(2)207 538 y Fn(V)24 b Fr(=)14 b(0,)i(then)h(it)g(is)f(true)g (that)h(T)l(r)o(\()p Fn(f)h Fr(+)10 b Fn(g)r Fr(\))k(=)g(T)l(r)7 b Fn(f)17 b Fr(+)11 b(T)l(r)d Fn(g)15 b Fr(=)f(0.)100 608 y(\(F)l(or)c Fn(f)17 b Fr(nilp)q(oten)o(t,)11 b(let)g(the)g(order)f (of)i Fn(f)17 b Fr(b)q(e)11 b(the)h(least)f Fn(n)g Fr(suc)o(h)f(that)h Fn(f)1363 590 y Fk(n)1405 608 y Fr(=)j(0.)d(After)g(the)h(date)f(of)g (this)0 677 y(lecture,)g(Professor)f(Bergmann)g(did)g(\014nd)h(a)g (coun)o(terexample,)f(where)h Fn(f)5 b Fr(,)12 b Fn(g)r Fr(,)e(and)h Fn(f)6 b Fr(+)q Fn(g)13 b Fr(eac)o(h)e(had)f(order)0 747 y(3.)j(He)g(conjectures)g(that)g(an)o(y)g(coun)o(terexample)e(will) i(ha)o(v)o(e)f(\(ord)c Fn(f)d Fr(\))1278 729 y Fj(\000)p Fl(1)1336 747 y Fr(+)t(\(ord)i Fn(g)r Fr(\))1523 729 y Fj(\000)p Fl(1)1581 747 y Fr(+)t(\(ord)g Fn(f)i Fr(+)t Fn(g)r Fr(\))1844 729 y Fj(\000)p Fl(1)1911 747 y Fm(\024)0 817 y Fr(1.\))0 946 y Fp(Uses)19 b(of)f(T)-5 b(ate's)19 b Fr(T)l(r)100 1016 y(W)l(e)13 b(consider)e Fn(V)25 b Fr(=)13 b Fn(K)520 1023 y Fk(P)554 1016 y Fr(.)g Fn(K)623 1023 y Fk(P)670 1016 y Fm(\033)734 1003 y Fr(^)722 1016 y Fn(O)760 1023 y Fk(P)806 1016 y Fr(with)927 1003 y(^)916 1016 y Fn(O)954 1023 y Fk(P)1000 1016 y Fr(a)g Fn(k)r Fr(-linear)d(subspace.)i(So)g(c)o(ho)q(ose)g Fn(\031)k Fr(:)e Fn(K)1768 1023 y Fk(P)1815 1016 y Fm(!)1890 1003 y Fr(^)1879 1016 y Fn(O)1917 1023 y Fk(P)0 1086 y Fr(a)h Fn(k)r Fr(-linear)e(pro)s(jection.)h(De\014ne)g Fn(f)20 b Fr(=)13 b Fn(f)742 1093 y Fk(x;y)815 1086 y Fm(2)h Fr(End)o(\()p Fn(V)e Fr(\))j(b)o(y)g Fn(f)5 b Fr(\()p Fn(v)r Fr(\))15 b(=)e Fn(\031)r Fr(\()p Fn(xy)r(v)r Fr(\))8 b Fm(\000)g Fn(y)r(\031)r Fr(\()p Fn(xv)r Fr(\))18 b(for)c Fn(v)i Fm(2)e Fn(V)d Fr(.)k Fn(f)21 b Fr(is)0 1162 y(\014nite-p)q(oten) o(t)13 b(as)g(dim)423 1169 y Fk(k)456 1162 y Fn(f)485 1144 y Fl(2)508 1162 y Fn(V)25 b(<)14 b Fm(1)p Fr(:)f Fn(f)5 b(V)26 b Fm(\032)840 1149 y Fr(^)828 1162 y Fn(O)866 1169 y Fk(p)894 1162 y Fr(+)5 b Fn(y)976 1149 y Fr(^)964 1162 y Fn(O)1002 1169 y Fk(p)1039 1162 y Fr(=)23 b(^)-34 b Fn(m)1136 1144 y Fk(a)1136 1176 y(P)1183 1162 y Fr(for)13 b(some)g Fn(a)p Fr(.)g(k)o(er)8 b Fn(f)20 b Fm(\033)14 b Fr(\()1633 1149 y(^)1622 1162 y Fn(O)1660 1169 y Fk(p)1697 1162 y Fr(:)f Fn(xy)r Fr(\))5 b Fm(\\)g Fr(\()1873 1149
y(^)1859 1162 y Fn(O)1897 1169 y Fk(p)1936 1162 y Fr(:)0 1233 y Fn(x)p Fr(\))21 b Fm(\033)29 b Fr(^)-34 b Fn(m)171 1215 y Fk(b)171 1247 y(P)224 1233 y Fr(for)20 b(some)f Fn(b)p Fr(.)i(If)g Fn(b)f Fm(\025)g Fn(a)p Fr(,)g(then)h Fn(f)849 1215 y Fl(2)892 1233 y Fr(=)e(0)i(as)f Fn(f)5 b(V)32 b Fm(\032)19 b Fr(k)o(er)8 b Fn(f)d Fr(.)21 b(Otherwise,)e Fn(f)26 b Fr(descends)19 b(to)i(a)0 1304 y(surjectiv)o(e)16 b(map)25 b(^)-34 b Fn(m)378 1286 y Fk(a)378 1318 y(P)411 1304 y Fn(=)9 b Fr(^)-34 b Fn(m)480 1286 y Fk(b)480 1318 y(P)527 1304 y Fm(!)13 b Fn(f)619 1286 y Fl(2)642 1304 y Fn(V)e Fr(,)17 b(and)f(dim)893 1311 y Fk(k)935 1304 y Fr(^)-34 b Fn(m)970 1286 y Fk(a)970 1318 y(P)1002 1304 y Fn(=)10 b Fr(^)-35 b Fn(m)1071 1286 y Fk(b)1071 1318 y(P)1118 1304 y Fr(=)14 b(deg)8 b Fn(P)f Fr(\()p Fn(a)12 b Fm(\000)f Fn(b)p Fr(\))k Fn(<)e Fm(1)p Fr(.)0 1434 y Fp(Tw)n(o)19 b(exercises)i(on)d Fn(f)100 1503 y Fr(\(1\))e(T)l(r)8 b Fn(f)263 1510 y Fk(x;y)337 1503 y Fr(is)16 b(indep)q(enden)o(t)e(of)i (the)h(c)o(hoice)e(of)h Fn(\031)r Fr(.)g(Use)g(Prop)q(ert)o(y)f(\(2\))i (of)f(T)l(r)f(and)g(the)h(fact)h(that)0 1573 y Fn(\031)28 1580 y Fl(1)62 1573 y Fm(\000)11 b Fn(\031)140 1580 y Fl(2)179 1573 y Fr(is)16 b(nilp)q(oten)o(t,)g(hence)g(\014nite-p)q (oten)o(t.)100 1645 y(\(2\))k(If)248 1632 y(^)236 1645 y Fn(O)274 1652 y Fk(P)327 1645 y Fr(=)f Fn(k)r Fr([[)p Fn(t)p Fr(]],)f(then)i Fn(K)678 1652 y Fk(P)731 1645 y Fr(=)f Fn(k)r Fr(\(\()p Fn(t)p Fr(\)\).)h(F)l(or)f Fn(x)h Fr(=)1144 1608 y Fh(P)1196 1660 y Fk(i)1221 1645 y Fn(a)1247 1652 y Fk(i)1264 1645 y Fn(t)1282 1627 y Fk(i)1299 1645 y Fr(,)f Fn(y)j Fr(=)1436 1608 y Fh(P)1489 1660 y Fk(j)1518 1645 y Fn(b)1539 1652 y Fk(j)1560 1645 y Fn(t)1578 1627 y Fk(j)1599 1645 y Fr(,)e(then)g(T)l(r)7 b Fn(f)1833 1652 y Fk(x;y)1911 1645 y Fr(=)0 1685 y Fh(P)53 1738 y Fk(i)p Fl(+)p Fk(j)r Fl(=0)177 1723 y Fn(j)s(a)227 1730 y Fk(i)243 1723 y Fn(b)264 1730 y Fk(j)286 1723 y Fr(.)16 b(Use)h Fn(\031)r Fr(\()461 1685 y Fh(P)514 1738 y Fk(i)539 1723 y Fn(c)561 1730 y Fk(i)577 1723 y Fn(t)595 1705 y Fk(i)612 1723 y Fr(\))d(=)697 1685 y Fh(P)750 1738 y Fk(i)p Fj(\025)p Fl(0)826 1723 y Fn(c)848 1730 y Fk(i)864 1723 y Fn(t)882 1730 y Fk(i)899 1723 y Fr(.)0 1853 y Fp(Lo)r(cal)20 b(Inner)e(Pro)r(duct)100 1923 y Fr(No)o(w)e(w)o(e)g(can)g(de\014ne)f(a)h(lo)q(cal)g(inner)g(pro) q(duct)f Fm(h)p Fn(x;)8 b(y)r Fm(i)1106 1930 y Fk(P)1155 1923 y Fr(=)13 b(T)l(r)8 b Fn(f)1291 1930 y Fk(x;y)1350 1923 y Fr(;)16 b(b)o(y)g(the)g(exercises,)g Fm(h)p Fn(x;)8 b(y)r Fm(i)1866 1930 y Fk(P)1917 1923 y Fr(is)0 1993 y(w)o(ell)16 b(de\014ned)g(and)f(agrees)h(with)h(res)694 2000 y Fk(P)727 1993 y Fr(\()p Fn(x)774 2000 y Fk(P)808 1993 y Fn(dy)r Fr(\))g(in)g(the)f(classical)g(case.)100 2063 y(Next)i(time)f(w)o(e)f(will)h(put)g(the)g(the)g(lo)q(cal)g(inner) f(pro)q(ducts)g(together)h(to)g(de\014ne)g Fn(D)q Fr(,)g(and)g(giv)o(e) f(an)0 2132 y(o)o(v)o(erview)g(of)h(T)l(ate's)f(pro)q(of)g(of)h(the)g (theorem.)0 2262 y Fp(Lecture)j(29)c Fr(\(No)o(v)o(em)o(b)q(er)g(3,)g (1995\).)0 2331 y(Scrib)q(e:)g(W)l(a)o(yne)g(Whitney)100 2461 y(So)h(last)g(time)g(w)o(e)g(w)o(ere)g(ab)q(out)g(to)h(get)f(to)h (T)l(ate's)f(pro)q(of)g(of)g(the)h(theorem)f(relating)f(\012)1757 2470 y Fk(K=k)1853 2461 y Fr(with)0 2531 y(Di\013)85 2540 y Fk(K=k)164 2531 y Fr(,)k(using)g(the)h(trace)g(of)h(\014nitep)q (oten)o(t)e(maps)g(T)l(ate)h(de\014ned.)f(As)h(usual,)f(let)h Fn(K)k Fm(\033)c Fn(k)i Fr(b)q(e)e(the)0 2600 y(function)16 b(\014eld)g(of)h(a)g(curv)o(e)f(o)o(v)o(er)f Fn(k)r Fr(,)h(that)i(is)e
Fn(K)i Fm(\033)13 b Fn(k)18 b Fr(is)f(\014nitely)f(generated)g(and)g (has)g(transcendence)0 2670 y(degree)g(1.)g(W)l(e)h(also)f(require)g (to)q(da)o(y)h(that)g Fn(K)g Fm(\033)d Fn(k)k Fr(is)e(separably)f (generated.)950 2790 y(60)p eop %%Page: 61 61 61 60 bop 0 50 a Fp(Ov)n(erview)21 b(of)e(the)g(Pro)r(of)100 123 y Fr(Let)h Fn(P)25 b Fm(2)19 b Fn(C)t Fr(,)g Fn(P)26 b Fr(a)20 b(closed)f(p)q(oin)o(t.)g(Then)918 111 y(^)907 123 y Fn(O)945 130 y Fk(p)986 123 y Fm(\032)g Fn(K)1086 130 y Fk(p)1129 123 y Fr(and)f(is)h(a)h Fn(k)r Fr(-linear)d(subspace)h (of)i Fn(K)1824 130 y Fk(p)1847 123 y Fr(.)g(W)l(e)0 200 y(mak)o(e)h(a)h(c)o(hoice)f Fn(\031)r Fr(:)h Fn(K)443 207 y Fk(p)489 200 y Fm(!)572 187 y Fr(^)561 200 y Fn(O)599 207 y Fk(p)644 200 y Fr(whic)o(h)e(is)i(a)f Fn(k)r Fr(-linear)f(pro)s (jection.)h(Then)g(for)g Fn(x)p Fr(,)h Fn(y)j Fm(2)e Fn(K)1764 207 y Fk(p)1787 200 y Fr(,)f(de\014ne)0 269 y Fn(f)24 276 y Fk(x;y)101 269 y Fr(:)d Fn(K)176 276 y Fk(p)218 269 y Fm(!)f Fn(K)328 276 y Fk(p)371 269 y Fr(b)o(y)h Fn(f)467 276 y Fk(x;y)526 269 y Fr(\()p Fn(v)r Fr(\))g(=)f Fn(\031)r Fr(\()p Fn(xy)r(v)r Fr(\))d Fm(\000)e Fn(y)r(\031)r Fr(\()p Fn(xv)r Fr(\))21 b(for)e Fn(v)h Fm(2)f Fn(K)1267 276 y Fk(p)1290 269 y Fr(.)h(W)l(e)f(sa)o(w)g(that)h Fn(f)1646 276 y Fk(x;y)1724 269 y Fr(w)o(as)f(\014nite-)0 339 y(p)q(oten)o(t,)13 b(and)g(T)l(r)7 b Fn(f)345 346 y Fk(x;y)418 339 y Fr(w)o(as)12 b(indep)q(enden)o(t)g(of)h(the)h(c)o (hoice)f(of)g Fn(\031)r Fr(,)g(so)g(w)o(e)g(could)g(de\014ne)g Fm(h)p Fn(x;)8 b(y)r Fm(i)1693 346 y Fk(P)1741 339 y Fr(=)14 b(T)l(r)7 b Fn(f)1877 346 y Fk(x;y)1936 339 y Fr(.)100 413 y(No)o(w)18 b(one)g(needs)g(to)g(c)o(hec)o(k)g(that)h Fm(h\001)p Fn(;)8 b Fm(\001i)842 420 y Fk(P)893 413 y Fr(:)16 b Fn(K)965 420 y Fk(p)1001 413 y Fm(\002)c Fn(K)1094 420 y Fk(p)1134 413 y Fm(!)17 b Fn(K)1243 420 y Fk(p)1285 413 y Fr(is)g Fn(k)r Fr(-bilinear,)f(and)i(that)h(one)f(has)0 482 y Fm(h)p Fn(x;)8 b(y)r(z)r Fm(i)139 489 y Fk(P)190 482 y Fr(=)16 b Fm(h)p Fn(xy)r(;)8 b(z)r Fm(i)384 489 y Fk(P)431 482 y Fr(+)j Fm(h)p Fn(xz)r(;)d(y)r Fm(i)620 489 y Fk(P)656 482 y Fr(.)17 b(Then)h(w)o(e)f(can)h(de\014ne)f(a)h (deriv)m(ation)f Fn(D)g Fr(:)f Fn(K)k Fm(!)c Fr(Di\013)1710 491 y Fk(K=k)1806 482 y Fr(b)o(y:)h(for)0 552 y Fn(y)f Fm(2)e Fn(K)21 b Fr(and)15 b Fn(x)g Fr(=)e(\()p Fn(x)388 559 y Fk(P)422 552 y Fr(\))i Fm(2)f Fp(A)546 559 y Fk(K)584 552 y Fr(,)719 643 y Fn(D)q Fr(\()p Fn(y)r Fr(\)\()p Fn(x)p Fr(\))j(=)969 596 y Fh(X)960 703 y Fk(P)5 b Fj(2)p Fk(C)1049 643 y Fm(h)p Fn(x)1096 650 y Fk(P)1130 643 y Fn(;)j(y)r Fm(i)1197 650 y Fk(P)0 818 y Fr(The)17 b(sum)e(ab)q(o)o(v) o(e)h(is)h(a)f(\014nite)h(sum:)e(if)i Fn(x;)8 b(y)17 b Fm(2)883 806 y Fr(^)872 818 y Fn(O)910 825 y Fk(P)943 818 y Fr(,)g(then)f Fn(f)5 b Fr(\()p Fn(K)1177 825 y Fk(P)1212 818 y Fr(\))14 b Fm(\032)1309 806 y Fr(^)1298 818 y Fn(O)1336 825 y Fk(P)1386 818 y Fr(and)i Fn(f)5 b Fr(\()1543 806 y(^)1531 818 y Fn(O)1569 825 y Fk(P)1603 818 y Fr(\))15 b(=)e(0,)k(so)f Fn(f)1835 800 y Fl(2)1872 818 y Fr(=)e(0)0 888 y(and)i(T)l(r)7 b Fn(f)20 b Fr(=)14 b(0.)i(Th)o(us)f(for)h(all)h(but)f(\014nitely)g(man)o(y)g Fn(P)7 b Fr(,)16 b Fm(h)p Fn(x)1090 895 y Fk(P)1124 888 y Fn(;)8 b(y)r Fm(i)1191 895 y Fk(P)1239 888 y Fr(=)14 b(0.)100 962 y(No)o(w)23 b(one)g(needs)f(to)i(c)o(hec)o(k)f(that)g Fn(D)q Fr(\()p Fn(y)r Fr(\))k(:)d Fp(A)997 969 y Fk(K)1061 962 y Fm(!)h Fn(k)g Fr(is)d(con)o(tinous,)g(and)g(that)i(the)f(residue)
0 1031 y(theorem)16 b(holds,)f(so)h(that)h Fn(K)h Fm(\032)13 b Fr(k)o(er)8 b Fn(D)q Fr(.)17 b(These)g(details)f(ma)o(y)f(b)q(e)i (found)f(in)g(T)l(ate's)g(pap)q(er.)100 1105 y(Then)i Fn(D)j Fr(is)d Fn(k)r Fr(-linear,)f(once)i(y)o(ou)g(realize)g(that)g (the)g Fn(k)r Fr(-v)o(ector)f(space)h(structure)f(on)h(Di\013)1819 1114 y Fk(K=k)1917 1105 y Fr(is)0 1175 y(giv)o(en)14 b(b)o(y)g Fn(z)r(\025)p Fr(\()p Fn(x)p Fr(\))h(=)f Fn(\025)p Fr(\()p Fn(z)r(x)p Fr(\))h(for)f Fn(\025)g Fm(2)g Fr(Di\013)767 1184 y Fk(K=k)846 1175 y Fr(,)g Fn(x)g Fm(2)g Fp(A)1006 1182 y Fk(K)1059 1175 y Fr(and)g Fn(z)i Fm(2)e Fn(k)r Fr(.)g(Also,)f Fn(D)q Fr(\()p Fn(y)r(z)r Fr(\))j(=)e Fn(y)r(D)q Fr(\()p Fn(z)r Fr(\))6 b(+)g Fn(z)r(D)q Fr(\()q Fn(y)r Fr(\).)100 1248 y(Th)o(us)16 b Fn(D)i Fr(is)f Fn(k)r Fr(-deriv)m(ation)f(of)i Fn(K)t Fr(.)f(So)g Fn(D)h Fr(giv)o(es)f(rise)g(to)g(a)h Fn(K)t Fr(-linear)d(map)i Fn(c)d Fr(:)h(\012)1614 1257 y Fk(K=k)1708 1248 y Fm(!)g Fr(Di\013)1857 1257 y Fk(K=k)1936 1248 y Fr(,)0 1318 y(suc)o(h)f(that)i Fn(D)f Fr(=)f Fn(cd)p Fr(,)g(where)h Fn(d)f Fr(:)f Fn(K)18 b Fm(!)c Fr(\012)773 1327 y Fk(K=k)867 1318 y Fr(is)g(the)i(natural)e(map.)g Fn(c)h Fr(is)g(the)g(map)f (desired)g(for)h(T)l(ate's)0 1388 y(theorem.)0 1536 y Fp(Extensions)k(of)f(F)-5 b(unction)20 b(Fields)h(of)d(Curv)n(es)100 1609 y Fr(W)l(e)e(no)o(w)f(mo)o(v)o(e)g(to)h(a)g(new)g(topic.)g (Consider)e(a)i(function)g(\014eld)f(of)h(a)g(curv)o(e)g(o)o(v)o(er)f Fn(k)r Fr(,)g Fn(K)j Fm(\033)13 b Fn(k)r Fr(,)j(and)0 1679 y(a)g(\014nite)f(extension)h Fn(L)d Fm(\033)h Fn(K)t Fr(.)h(Then)h Fn(L)f Fr(itself)h(is)g(a)f(function)g(\014eld)h(of)g(a)f (curv)o(e.)g(The)h(inclusion)e Fn(K)k Fm(\032)13 b Fn(L)0 1749 y Fr(corresp)q(onds)h(to)j(a)g(dominan)o(t)e(\(non-constan)o(t)g (in)h(this)g(case\))h(morphism)d Fn(C)1465 1756 y Fk(L)1508 1749 y Fm(!)g Fn(C)1608 1756 y Fk(K)1646 1749 y Fr(.)100 1822 y(Note)19 b(that)f(if)h Fn(k)g Fr(is)f(the)h(exact)g(\014eld)e(of) i(constan)o(ts)e(of)h Fn(K)t Fr(,)g(that)h(is)f Fn(K)e Fm(\\)1490 1809 y Fr(\026)1488 1822 y Fn(k)i Fr(=)f Fn(k)r Fr(,)g(it)i(ma)o(y)f(not)g(b)q(e)0 1892 y(the)e(case)f(that)h Fn(k)h Fr(is)f(the)f(exact)i(\014eld)e(of)h(constan)o(ts)e(of)i Fn(L)p Fr(.)g(That)f(is)g(to)h(sa)o(y)l(,)f(w)o(e)g(will)h(ha)o(v)o(e)f (to)g(consider)0 1962 y(the)20 b(situation)f Fn(L)14 b Fm(\\)395 1948 y Fr(\026)394 1962 y Fn(k)21 b Fm(6)p Fr(=)e Fn(k)r Fr(.)g(So)h(previous)f(results)f(whic)o(h)h(assumed)f Fn(K)g Fm(\\)1452 1948 y Fr(\026)1451 1962 y Fn(k)j Fr(=)e Fn(k)j Fr(ma)o(y)d(need)h(to)g(b)q(e)0 2031 y(reform)o(ulated.)100 2105 y(So)14 b(for)h(an)o(y)f(function)h(\014eld)f(of)h(a)g(curv)o(e)f Fn(K)k Fm(\033)c Fn(k)r Fr(,)g(set)h Fn(k)1131 2112 y Fl(0)1167 2105 y Fr(=)f Fn(K)d Fm(\\)1316 2092 y Fr(\026)1314 2105 y Fn(k)r Fr(,)k(the)g(exact)h(\014eld)e(of)h(constan)o(ts)0 2174 y(of)i Fn(K)t Fr(.)f(Then)g Fn(k)289 2181 y Fk(o)325 2174 y Fm(\033)e Fn(k)r Fr(,)i(and)f([)p Fn(k)572 2181 y Fl(0)608 2174 y Fr(:)f Fn(k)r Fr(])f Fn(<)h Fm(1)p Fr(.)100 2248 y(No)o(w)k(consider)f(the)h(statemen)o(t)h(of)f (Riemann-Ro)q(c)o(h.)e(All)i(dimensions)e(in)o(v)o(olv)o(ed)h(are)h (prop)q(erly)0 2318 y(o)o(v)o(er)13 b Fn(k)130 2325 y Fl(0)152 2318 y Fr(.)h(But)h(if)f(w)o(e)g(m)o(ultiply)f(through)f(b)o (y)i([)p Fn(k)880 2325 y Fl(0)916 2318 y Fr(:)g Fn(k)r Fr(],)f(all)h(dimensions)d(will)j(b)q(e)g(o)o(v)o(er)f Fn(k)r Fr(.)h(This)f(will)h(ha)o(v)o(e)0 2387 y(the)j(e\013ect)g(of)g (c)o(hanging)e(the)i(1)f(in)g(Riemann-Ro)q(c)o(h)e(to)j([)p Fn(k)1108 2394 y Fl(0)1144 2387 y Fr(:)d Fn(k)r Fr(].)100 2461 y(With)k(these)f(preliminaries,)e(w)o(e)j(can)f(again)g(consider)g
(an)g(extension)h(of)g(functions)f(\014elds)f Fn(L)g Fm(\033)0 2531 y Fn(K)i Fm(\033)13 b Fn(k)r Fr(.)g(Ultimately)h(w)o(e)f (will)g(pro)o(v)o(e)f(the)i(Hurwitz)f(form)o(ula,)f(also)h(kno)o(wn)f (as)h(the)h(Riemann-Hurwitz)0 2600 y(form)o(ula)19 b(or)h(the)h (Hurwitz-Zeuthen)f(form)o(ula.)f(This)h(form)o(ula)f(links)h Fn(g)1384 2607 y Fk(L)1414 2600 y Fr(,)g(the)h(gen)o(us)f(of)g Fn(L)p Fr(,)h(to)g Fn(g)1898 2607 y Fk(K)1936 2600 y Fr(.)0 2670 y(The)16 b(pro)q(of)g(of)h(this)f(form)o(ula)f(will)g(come) i(from)e(a)i(consideration)d(of)j(the)f(map)g Fn(D)i Fr(from)e(the)g(previous)950 2790 y(61)p eop %%Page: 62 62 62 61 bop 0 50 a Fr(section,)16 b(no)o(w)g(indexed)g(b)o(y)g(the)h (\014eld)f(in)g(question.)g(W)l(e)h(will)f(sho)o(w)f(that:)630 181 y Fn(D)671 188 y Fk(L)701 181 y Fr(\()p Fn(y)r Fr(\)\()p Fn(x)p Fr(\))h(=)d Fn(D)940 188 y Fk(K)979 181 y Fr(\()p Fn(y)r Fr(\)\(T)l(r)1115 190 y Fg(A)1149 195 y Fc(L)1175 190 y Fk(=)p Fg(A)1229 195 y Fc(K)1272 181 y Fn(x)p Fr(\))0 313 y(where)h Fn(y)j Fm(2)d Fn(K)19 b Fr(and)14 b Fn(x)g Fm(2)g Fp(A)518 320 y Fk(L)548 313 y Fr(.)h(Of)g(course,)f(to)h(do)g (this,)g(w)o(e)f(will)h(need)g(to)g(de\014ne)f(the)h(trace)h(T)l(r)1787 322 y Fg(A)1821 327 y Fc(L)1847 322 y Fk(=)p Fg(A)1901 327 y Fc(K)1936 313 y Fr(.)0 446 y Fp(A)j(Basis-F)-5 b(ree)20 b(De\014nition)g(of)e Fr(T)l(r)100 516 y(Let)d Fn(A)g Fr(b)q(e)g(a)g(ring,)f(and)g(let)h Fn(B)i Fr(b)q(e)e(an)g Fn(A)p Fr(-algebra)e(whic)o(h)h(is)g(a)h(\014nite,)f(free)h Fn(A)p Fr(-mo)q(dule)f(\(or)h(lo)q(cally)0 586 y(free\).)21 b(Then)e(w)o(e)i(will)f(construct)f(a)i(map)e(T)l(r)876 595 y Fk(B)r(=)n(A)978 586 y Fr(:)h Fn(B)j Fm(!)d Fn(A)g Fr(whic)o(h)g(is)g(giv)o(en)g(b)o(y)g(T)l(r)1659 595 y Fk(B)r(=)n(A)1741 586 y Fr(\()p Fn(b)p Fr(\))i(=)e(the)0 656 y(trace)d(of)f(the)h(the)g Fn(A)p Fr(-linear)e(map)h Fn(B)g Fm(!)e Fn(B)19 b Fr(giv)o(en)d(b)o(y)g Fn(x)e Fm(7!)g Fn(bx)p Fr(.)100 726 y(So)j(w)o(e)g(ha)o(v)o(e)g(the)g(natural) g(map)f Fn(B)i Fm(!)d Fr(Hom)958 733 y Fk(A)990 726 y Fr(\()p Fn(B)r(;)8 b(B)r Fr(\))19 b(giv)o(en)e(b)o(y)g Fn(b)f Fm(7!)f Fr(\()p Fn(x)h Fm(7!)f Fn(bx)p Fr(\).)k(No)o(w)e(for)h (an)o(y)0 796 y(t)o(w)o(o)d(A-mo)q(dules)f Fn(C)k Fr(and)d Fn(D)q Fr(,)h(there)f(is)g(a)g(natural)f(map)g(Hom)1164 803 y Fk(A)1197 796 y Fr(\()p Fn(D)q(;)8 b(A)p Fr(\))i Fm(\012)1385 803 y Fk(A)1426 796 y Fn(C)17 b Fm(!)c Fr(Hom)1646 803 y Fk(A)1679 796 y Fr(\()p Fn(D)q(;)8 b(C)t Fr(\))16 b(giv)o(en)0 866 y(b)o(y)e Fn(f)e Fm(\012)6 b Fn(c)13 b Fm(7!)h Fr(\()p Fn(d)g Fm(7!)f Fn(f)5 b Fr(\()p Fn(d)p Fr(\))p Fn(c)p Fr(\).)16 b(When)d Fn(C)k Fr(=)d Fn(A)p Fr(,)g(this)g(map)f(is)h(eviden)o(tly)g(an)g(isomorphism)o(,)d(and)j (the)g(map)0 935 y(resp)q(ects)i Fm(\010)p Fr(,)g(so)g(it)h(is)f(an)g (isomorphism)e(when)i Fn(B)965 922 y Fm(\030)965 938 y Fr(=)1017 935 y Fn(A)1054 917 y Fk(n)1098 935 y Fr(\(or)h(if)f Fn(B)j Fr(is)d(a)h(direct)f(summand)e(of)j(a)f(\014nite)0 1005 y(free)h(mo)q(dule,)e(that)i(is)f(a)h(\014nite)f(pro)s(jectiv)o(e) g(mo)q(dule,)g(equiv)m(alen)o(tly)h(a)f(\014nite)g(lo)q(cally)h(free)g (mo)q(dule\).)100 1076 y(Th)o(us)c(w)o(e)i(ha)o(v)o(e)f(sho)o(wn)f(Hom) 662 1083 y Fk(A)695 1076 y Fr(\()p Fn(B)r(;)8 b(B)r Fr(\))850 1062 y Fm(\030)850 1078 y Fr(=)903 1076 y(Hom)1006 1083 y Fk(A)1039 1076 y Fr(\()p Fn(B)r(;)g(A)p Fr(\))g Fm(\012)1223 1083 y Fk(A)1264 1076 y Fn(B)r Fr(.)15 b(Eviden)o(tly)l(,)f(w)o(e)h(ha) o(v)o(e)f(a)h(natural)0 1145 y(map)j(Hom)217 1152 y Fk(A)250 1145 y Fr(\()p Fn(B)r(;)8 b(A)p Fr(\))14 b Fm(\012)440 1152 y Fk(A)485 1145 y Fn(B)21 b Fm(!)d Fn(A)p Fr(:)i(just)f
Fn(f)f Fm(\012)13 b Fn(b)19 b Fm(7!)f Fn(f)5 b Fr(\()p Fn(b)p Fr(\).)21 b(Comp)q(osing)d(the)h(isomorphism)d(with)j(this)0 1215 y(map,)d(w)o(e)g(get)h Fn(T)7 b(r)339 1224 y Fk(B)r(=)n(A)436 1215 y Fr(:)13 b Fn(B)k Fm(!)c Fn(A)p Fr(.)0 1348 y Fp(Bac)n(k)20 b(to)f(the)g(Adeles)100 1418 y Fr(So)14 b(to)i(de\014ne)f(T)l(r)421 1427 y Fg(A)455 1432 y Fc(L)481 1427 y Fk(=)p Fg(A)535 1432 y Fc(K)570 1418 y Fr(,)g(it)h(su\016ces)e(to)h(sho)o(w)f(that)i Fp(A)1147 1425 y Fk(L)1192 1418 y Fr(is)f(a)g(\014nite)g(free)g Fp(A)1544 1425 y Fk(K)1598 1418 y Fr(mo)q(dule.)g(T)l(o)g(that)0 1488 y(end:)0 1621 y Fp(Theorem)f Fr(If)i Fn(L)d Fm(\033)h Fn(K)k Fm(\033)13 b Fn(k)18 b Fr(with)d Fn(L)f Fm(\033)f Fn(K)20 b Fr(\014nite)15 b(and)g Fn(K)j Fm(\033)c Fn(k)j Fr(\014nitely)e(generated)g(of)h(transcendence)0 1691 y(degree)g(1,)g(then)h(the)g(natural)e(map)h Fn(L)11 b Fm(\012)779 1698 y Fk(K)828 1691 y Fp(A)871 1698 y Fk(K)924 1691 y Fm(!)i Fp(A)1030 1698 y Fk(L)1077 1691 y Fr(is)j(an)g(isomorphism.)100 1761 y Fp(Pro)r(of)g Fr(First)h(w)o(e)h(do)f(ha)o(v)o(e)h(a)g(natural)f(map:)g Fn(L)f Fm(\032)g Fp(A)1135 1768 y Fk(L)1183 1761 y Fr(in)i(a)g(natural) f(w)o(a)o(y)l(,)g(and)h(as)f Fn(K)1777 1768 y Fk(P)1827 1761 y Fm(\032)f Fn(L)1916 1768 y Fk(Q)0 1831 y Fr(for)g(ev)o(ery)h Fn(Q)d Fm(7!)g Fn(P)7 b Fr(,)17 b(w)o(e)f(ha)o(v)o(e)g(a)h(map)f Fp(A)779 1838 y Fk(K)831 1831 y Fm(!)e Fp(A)938 1838 y Fk(L)985 1831 y Fr(giv)o(en)i(b)o(y)g(\()p Fn(x)1231 1838 y Fk(P)1266 1831 y Fr(\))e Fm(7!)g Fr(\()p Fn(y)1406 1838 y Fk(Q)1440 1831 y Fr(\))k(where)e(if)g Fn(Q)f Fm(7!)f Fn(P)7 b Fr(,)16 b(then)0 1901 y Fn(y)24 1908 y Fk(Q)72 1901 y Fr(=)e Fn(x)153 1908 y Fk(P)186 1901 y Fr(.)j(Hence)g(w)o(e)f (ha)o(v)o(e)g(a)h(natural)e(map)h Fn(L)11 b Fm(\012)964 1908 y Fk(K)1013 1901 y Fp(A)1056 1908 y Fk(K)1108 1901 y Fm(!)j Fp(A)1215 1908 y Fk(L)1245 1901 y Fr(.)100 1971 y(Next)f(times)e(w)o(e)h(will)f(giv)o(e)h(a)f(\\funn)o(y")g(pro)q(of)h (using)e(divisors)g(that)i(this)g(map)f(is)g(an)g(isomorphism.)0 2104 y Fp(Lecture)20 b(30)c Fr(\(No)o(v)o(em)o(b)q(er)g(6,)g(1995\).)0 2174 y(Scrib)q(e:)g(Nic)o(k)h(Mey)o(er)100 2308 y(As)h(alw)o(a)o(ys,)f (let)i Fn(k)f Fm(\032)e Fn(K)22 b Fr(b)q(e)d(a)f(\014nitely)g (generated)g(\014eld)g(extension)g(of)g(transcendence)f(degree)0 2378 y(1.)f(Let)i Fn(K)f Fm(\032)d Fn(L)i Fr(b)q(e)h(a)g(\014nite)f (\014eld)g(extension.)100 2448 y(Then)i(the)g(inclusion)f Fn(K)k Fm(!)c Fn(L)i Fr(induces)e(the)i(map)f Fn(C)1128 2455 y Fk(L)1174 2448 y Fm(!)f Fn(C)1277 2455 y Fk(K)1315 2448 y Fr(.)i(F)l(or)e(p)q(oin)o(t)h Fn(Q)f Fm(2)h Fn(C)1714 2455 y Fk(L)1762 2448 y Fr(mapping)0 2524 y(to)i(p)q(oin)o(t)f Fn(P)25 b Fm(2)19 b Fn(C)342 2531 y Fk(K)380 2524 y Fr(,)g(w)o(e)g(ha)o (v)o(e)g Fm(O)647 2531 y Fk(p)688 2524 y Fr(=)g Fm(O)786 2531 y Fk(q)820 2524 y Fm(\\)13 b Fn(K)t Fr(.)20 b(W)l(e)f(recall)g (that)h Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\))19 b(=)g([)p Fn(L)1570 2506 y Fj(\003)1611 2524 y Fr(:)1662 2511 y(^)1643 2524 y Fm(O)1683 2531 y Fk(q)1705 2489 y Fj(\003)1741 2524 y Fm(\001)13 b Fn(K)1814 2506 y Fj(\003)1836 2524 y Fr(])20 b(and)0 2600 y Fn(f)5 b Fr(\()p Fn(Q=P)i Fr(\))17 b(=)e([)p Fn(k)r Fr(\()p Fn(Q)p Fr(\))h(:)f Fn(k)r Fr(\()p Fn(P)7 b Fr(\)],)17 b(and)g(note)g(that)892 2588 y(^)873 2600 y Fm(O)913 2607 y Fk(q)952 2600 y Fr(is)g(a)h(mo)q(dule)e(o)o(v)o (er)1349 2588 y(^)1330 2600 y Fm(O)1370 2607 y Fk(p)1410 2600 y Fr(free)h(of)h(rank)f Fn(e)12 b Fm(\001)f Fn(f)21 b Fr(=)15 b([)p Fn(L)1887 2607 y Fk(Q)1936 2600 y Fr(:)0 2670 y Fn(K)42 2677 y Fk(P)76 2670 y Fr(],)h(where)g Fn(L)298 2677 y Fk(Q)348 2670 y Fr(is)g(the)h(completion)f(of)g
Fn(L)h Fr(with)f(resp)q(ect)h(to)g Fn(Q)p Fr(.)g(W)l(e)f(see)h(that)g (if)g(w)o(e)f(\014x)h Fn(P)7 b Fr(,)16 b(then)950 2790 y(62)p eop %%Page: 63 63 63 62 bop 652 76 a Fh(X)637 183 y Fk(Q)p Fj(!)p Fk(P)747 123 y Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\))p Fn(f)e Fr(\()p Fn(Q=P)i Fr(\))17 b(=)c([)p Fn(L)h Fr(:)g Fn(K)t Fr(])p Fn(:)100 284 y Fr(Finally)l(,)h(w)o(e)h(note)h(that)743 364 y Fn(L)11 b Fm(\012)827 371 y Fk(K)876 364 y Fn(K)918 371 y Fk(P)966 364 y Fm($)1048 317 y Fh(Y)1029 424 y Fk(Q)p Fj(!)p Fk(P)1139 364 y Fn(L)1173 371 y Fk(Q)0 523 y Fr(de\014nes)16 b(an)g(isomorphism)d(of)k Fn(K)623 530 y Fk(P)656 523 y Fr(-algebras.)100 594 y(W)l(e)f(no)o(w)g(pro)q (ceed)g(with)h(a)f(theorem.)100 665 y Fp(Theorem)p Fr(:)c(The)j (natural)e(map)h Fn(L)7 b Fm(\012)814 672 y Fk(K)860 665 y Fp(A)903 672 y Fg(K)954 665 y Fm(!)14 b Fp(A)1061 672 y Fg(L)1105 665 y Fr(is)h(an)f(isomorphism)d(of)k Fp(A)1608 672 y Fg(K)1646 665 y Fr(-algebras.)d(I.e.:)0 741 y(if)17 b Fn(L)f Fr(is)g(giv)o(en)g(a)h(basis)e(-)i Fn(L)c Fr(=)574 704 y Fh(P)627 716 y Fl([)p Fk(L)p Fl(:)p Fk(K)r Fl(])627 756 y Fk(i)p Fl(=1)735 741 y Fn(K)i Fm(\001)10 b Fn(!)847 748 y Fk(i)881 741 y Fr(-)16 b(then)g(the)h Fn(!)1144 748 y Fk(i)1161 741 y Fr(,)f(view)o(ed)h(as)f(elemen)o(ts)g (of)g Fp(A)1719 748 y Fg(L)1749 741 y Fr(,)h(form)e(an)0 811 y Fp(A)43 818 y Fg(K)100 811 y Fr(basis)j(for)h Fp(A)347 818 y Fg(L)377 811 y Fr(.)g(\(Note:)h(the)f Fn(!)685 818 y Fk(i)721 811 y Fr(form)g(a)g Fn(K)928 818 y Fk(P)980 811 y Fr(basis)f(for)1184 774 y Fh(Q)1231 826 y Fk(Q)p Fj(!)p Fk(P)1344 811 y Fn(L)1378 818 y Fk(Q)1412 811 y Fr(,)h(where)f Fn(L)1625 818 y Fk(Q)1678 811 y Fr(is)h(de\014ned)f (as)0 885 y(ab)q(o)o(v)o(e\).)100 961 y Fp(Pro)r(of)p Fr(:)12 b(It)j(su\016ces)e(to)i(pro)o(v)o(e)e(that)h(for)g(almost)g (all)g Fn(P)7 b Fr(,)14 b(the)h Fn(!)1264 968 y Fk(i)1295 961 y Fr(form)e(a)i(basis)e(of)1624 923 y Fh(Q)1671 976 y Fk(Q)p Fj(!)p Fk(P)1802 948 y Fr(^)1784 961 y Fm(O)1824 968 y Fk(q)1860 961 y Fr(o)o(v)o(er)19 1032 y(^)0 1045 y Fm(O)40 1052 y Fk(p)63 1045 y Fr(,)j(since)482 1130 y Fn(L)11 b Fm(\012)566 1137 y Fk(K)615 1130 y Fp(A)658 1137 y Fg(K)710 1116 y Fm(\030)710 1132 y Fr(=)762 1130 y Fm(\010)801 1104 y Fl([)p Fk(L)p Fl(:)p Fk(K)r Fl(])801 1144 y Fk(i)p Fl(=1)901 1130 y Fp(A)944 1137 y Fg(K)993 1130 y Fm(\001)g Fn(!)1049 1137 y Fk(i)1079 1130 y Fr(=)1132 1082 y Fh(Y)1148 1189 y Fk(P)1204 1109 y Fj(0)1229 1130 y Fm(\010)1268 1137 y Fk(i)1295 1130 y Fn(K)1337 1137 y Fk(P)1382 1130 y Fm(\001)f Fn(!)1437 1137 y Fk(i)1454 1130 y Fn(;)0 1295 y Fr(\(where)16 b(the)h(restricted)f(pro)q(duct)g (is)g(tak)o(en)h(with)f(resp)q(ect)h(to)g Fm(\010)1228 1302 y Fk(i)1263 1282 y Fr(^)1244 1295 y Fm(O)1284 1302 y Fk(p)1318 1295 y Fm(\001)11 b Fn(!)1374 1302 y Fk(i)1390 1295 y Fr(\),)17 b(and)762 1440 y Fp(A)805 1447 y Fg(L)849 1440 y Fr(=)902 1392 y Fh(Y)918 1499 y Fk(P)974 1419 y Fj(0)1015 1392 y Fh(Y)996 1499 y Fk(Q)p Fj(!)p Fk(P)1106 1440 y Fn(L)1140 1447 y Fk(Q)1174 1440 y Fn(;)0 1636 y Fr(\(where)i(the)g(restricted)g(pro)q(duct)f(is)g(tak)o(en)i(with)e (resp)q(ect)i(to)1211 1599 y Fh(Q)1258 1651 y Fk(Q)p Fj(!)p Fk(P)1389 1624 y Fr(^)1371 1636 y Fm(O)1411 1643 y Fk(q)1432 1636 y Fr(\).)f(So)g(w)o(e)g(get)g(an)g(isomor-)0 1710 y(phism)c Fn(f)171 1717 y Fk(P)218 1710 y Fr(:)f Fm(\010)p Fn(K)327 1717 y Fk(P)371 1710 y Fm(\001)d Fn(!)427
1717 y Fk(i)458 1710 y Fm(!)521 1673 y Fh(Q)568 1725 y Fk(Q)p Fj(!)p Fk(P)681 1710 y Fn(L)715 1717 y Fk(Q)749 1710 y Fr(.)100 1795 y(No)o(w)20 b(all)g(w)o(e)g(need)f(to)i(do)f(is)g (sho)o(w)f(that)h(if)h Fn(y)h Fm(2)1065 1758 y Fh(Q)1112 1810 y Fk(Q)p Fj(!)p Fk(P)1243 1783 y Fr(^)1225 1795 y Fm(O)1265 1802 y Fk(q)1286 1795 y Fr(,)e Fn(y)i Fr(=)1425 1758 y Fh(P)1478 1810 y Fk(i)1503 1795 y Fn(\037)1534 1802 y Fk(i)1550 1795 y Fn(!)1581 1802 y Fk(i)1598 1795 y Fn(;)8 b Fr(\()p Fn(\037)1670 1802 y Fk(i)1707 1795 y Fm(2)20 b Fn(K)1802 1802 y Fk(P)1836 1795 y Fn(;)8 b(P)32 b(=)-30 b Fm(2)0 1879 y Fr(some)16 b(\014nite)g(set)h(dep)q (ending)e(only)i(on)f(the)h Fn(!)865 1886 y Fk(i)881 1879 y Fr(\),)g(then)f Fn(\037)1075 1886 y Fk(i)1106 1879 y Fm(2)1172 1867 y Fr(^)1153 1879 y Fm(O)1193 1886 y Fk(p)1216 1879 y Fr(.)g(\(I.e.,)h(the)g Fn(\037)1481 1886 y Fk(i)1514 1879 y Fr(are)f("in)o(tegral"\).)100 1950 y(T)l(o)f(sho)o(w)f(this,)h(w)o(e)g(do)g(something)f(unnatural)g (and)g(strange.)h(First,)f(w)o(e)h(view)h Fn(y)i Fr(as)d(an)g(elemen)o (t)0 2020 y(of)j Fp(A)101 2027 y Fg(L)149 2020 y Fr(and)g Fn(x)h Fr(as)f(an)g(elemen)o(t)f(of)i Fp(A)716 2027 y Fg(K)753 2020 y Fr(,)g(supplemen)o(ting)c(with)j(zeros)g(as)g (necessary)l(.)f(First,)g(w)o(e)h(\014nd)0 2090 y(divisor)h Fn(A)i Fr(=)284 2052 y Fh(P)336 2105 y Fk(P)366 2095 y Fi(0)391 2090 y Fn(a)417 2097 y Fk(P)447 2087 y Fi(0)464 2090 y Fn(P)503 2072 y Fj(0)538 2090 y Fr(of)f Fn(K)25 b Fr(suc)o(h)19 b(that)i Fn(K)d Fr(+)13 b Fp(a)21 b Fr(=)f Fp(A)1156 2097 y Fg(K)1194 2090 y Fr(,)g(where)g Fp(a)h Fr(=)1484 2052 y Fh(Q)1531 2105 y Fk(P)1561 2095 y Fi(0)1616 2088 y Fr(^)1586 2090 y Ff(m)1624 2097 y Fk(P)1654 2087 y Fi(0)1671 2065 y Fk(a)1693 2074 y Fc(P)1719 2067 y Fi(0)1737 2090 y Fr(;)g(no)o(w,)e(w)o(e)0 2159 y(\014nd)g(divisor)f Fn(B)j Fr(=)384 2122 y Fh(P)437 2174 y Fk(Q)468 2164 y Fi(0)492 2159 y Fn(b)513 2166 y Fk(Q)544 2157 y Fi(0)561 2159 y Fn(Q)600 2141 y Fj(0)634 2159 y Fr(of)e Fn(L)h Fr(\(w)o(e)f(will)g(assume,)f(WLOG,)i(that)f(the)h Fn(b)1542 2166 y Fk(Q)1573 2157 y Fi(0)1608 2159 y Fm(\025)f Fr(0\),)g(suc)o(h)g (that)0 2208 y Fh(P)53 2260 y Fk(i)77 2245 y Fp(a)11 b Fm(\001)g Fn(!)172 2252 y Fk(i)211 2245 y Fm(\022)22 b Fp(b)p Fr(,)g(where)f Fp(b)h Fr(=)604 2208 y Fh(Q)651 2260 y Fk(Q)682 2250 y Fi(0)737 2243 y Fr(^)706 2245 y Ff(m)744 2252 y Fk(Q)775 2242 y Fi(0)792 2220 y Fk(b)810 2230 y Fc(Q)837 2223 y Fi(0)855 2245 y Fr(;)f(\014nally)l(,)g(w)o(e)g (\014nd)g Fn(C)26 b Fr(=)1362 2208 y Fh(P)1414 2260 y Fk(P)1444 2250 y Fi(0)1469 2245 y Fn(c)1491 2252 y Fk(P)1521 2242 y Fi(0)1537 2245 y Fn(P)1576 2227 y Fj(0)1612 2245 y Fr(of)c Fn(K)j Fr(suc)o(h)c(that)0 2319 y Fn(K)c Fr(+)c Fp(a)19 b Fr(=)g Fp(A)259 2326 y Fg(K)297 2319 y Fr(,)h(where)f Fn(L)13 b Fm(\\)h Fp(b)20 b Fr(\(whic)o(h)f(is)g(\014nite)h(o)o(v)o(er) f Fn(k)r Fr(,)g(since)g Fn(l)q Fr(\()p Fm(\000)p Fn(B)r Fr(\))i(is)e(\014nite\))h(is)g(con)o(tained)e(in)0 2351 y Fh(P)53 2404 y Fk(i)77 2389 y Fp(c)p Fn(!)133 2396 y Fk(i)151 2389 y Fr(,)e(where)g Fp(c)e Fr(=)417 2351 y Fh(Q)464 2404 y Fk(P)494 2394 y Fi(0)549 2387 y Fr(^)519 2389 y Ff(m)557 2396 y Fk(P)587 2386 y Fi(0)604 2364 y Fk(c)622 2373 y Fc(P)648 2366 y Fi(0)666 2389 y Fr(.)100 2460 y Fp(Claim)p Fr(:)j(W)l(e)g(can)f(dra)o(w)f(the)i(conclusion)e(w)o (e)h(wish,)g(no)o(w,)g(if)g Fn(a)1296 2467 y Fk(P)1326 2457 y Fi(0)1357 2460 y Fm(\025)e Fr(0)p Fn(;)8 b(c)1479 2467 y Fk(P)1509 2457 y Fi(0)1539 2460 y Fm(\025)13 b
Fr(0.)100 2531 y Fp(Pro)r(of)p Fr(:)j Fn(y)j Fr(=)370 2493 y Fh(P)431 2531 y Fn(\037)462 2538 y Fk(i)478 2531 y Fn(!)509 2538 y Fk(i)543 2531 y Fm(2)e Fp(A)636 2538 y Fg(L)666 2531 y Fn(:)p Fp(A)723 2538 y Fg(K)777 2531 y Fm(3)g Fn(\037)858 2538 y Fk(i)892 2531 y Fr(=)f Fn(k)973 2538 y Fk(i)1002 2531 y Fr(+)c Fn(a)1079 2538 y Fk(i)1096 2531 y Fr(,)18 b(where)g Fn(k)1300 2538 y Fk(i)1334 2531 y Fm(2)f Fn(K)q(;)8 b(a)1475 2538 y Fk(i)1509 2531 y Fm(2)17 b(a)p Fr(.)i(No)o(w)f Fn(y)h Fr(=)e Fn(l)c Fr(+)f Fn(b)p Fr(,)0 2600 y(where)20 b Fn(l)i Fr(=)244 2563 y Fh(P)305 2600 y Fn(k)331 2607 y Fk(i)361 2600 y Fm(\001)14 b Fn(!)420 2607 y Fk(i)457 2600 y Fm(2)21 b Fn(L)p Fr(,)g Fn(b)g Fr(=)681 2563 y Fh(P)742 2600 y Fn(a)768 2607 y Fk(i)799 2600 y Fm(\001)14 b Fn(!)858 2607 y Fk(i)895 2600 y Fm(2)21 b Fp(b)p Fr(.)f(So)h Fn(y)i Fm(2)e Fp(b)p Fr(,)f(since)g Fn(y)j Fr(has)d(ev)o(erywhere)g(in)o(tegral)0 2670 y(co)q(ordinates;)h(th)o(us)g Fn(l)j Fm(2)f Fp(b)f Fm(\032)h Fn(L)14 b Fm(\\)h Fp(b)p Fr(.)22 b(This)f(yields,)g (\014nally)l(,)g(that)i Fn(k)1358 2677 y Fk(i)1397 2670 y Fm(2)g Fp(c)p Fr(,)f(whic)o(h)f(w)o(as)g(what)h(w)o(e)950 2790 y(63)p eop %%Page: 64 64 64 63 bop 0 50 a Fr(required.)18 b(The)h(pro)q(of)g(is)g(complete.)g (\(Note:)h(when)f(w)o(e)g(completed)g(\014elds)f(\014rst)h(b)q(efore)g (taking)h(the)0 120 y(pro)q(duct)c(in)g(our)g(de\014nition)f(of)i(the)g (adele)f(ring,)f(it)i(w)o(as)f(with)g(this)g(pro)q(of)h(in)f(mind\).) 100 190 y(On)i(to)i(another)e(topic!)i(Let)f Fn(\025)h Fr(b)q(e)f(a)h(W)l(eil)f(di\013eren)o(tial)f(of)h Fn(K)t Fr(;)g(i.e.,)g Fn(\025)f Fr(:)g Fp(A)1556 197 y Fg(K)1613 190 y Fm(!)g Fn(k)r Fr(,)g(with)i Fn(\025)f(k)r Fr(-)0 260 y(linear,)c(con)o(tin)o(uous,)e(surjectiv)o(e,)i(and)h(0)f(on)h Fn(K)t Fr(.)f(Then)h(w)o(e)f(asso)q(ciate)h(to)g Fn(\025)g Fr(the)g Fp(canonical)k(divisor)0 330 y Fn(W)47 337 y Fk(\025)73 330 y Fr(.)d(W)l(e)f(ha)o(v)o(e)g(that)h(deg)9 b Fn(W)543 337 y Fk(\025)583 330 y Fr(=)k(2)p Fn(g)684 337 y Fk(K)734 330 y Fm(\000)d Fr(2[)p Fn(K)15 b Fm(\\)925 317 y Fr(\026)923 330 y Fn(k)h Fr(:)d Fn(k)r Fr(].)100 400 y(Let's)f(lo)q(ok)g(at)h(the)f(dual)f(space)h(Hom)798 407 y Fk(K)836 400 y Fr(\()p Fn(L;)c(K)t Fr(\).)13 b(This)e(is)h(a)g(v) o(ector)g(space)g(o)o(v)o(er)f(K)h(with)g(dimension)0 470 y([)p Fn(L)18 b Fr(:)g Fn(K)t Fr(].)g(It's)h(also)g(an)f Fn(L)p Fr(-v)o(ector)h(space;)f(for)h Fn(c)f Fm(2)g Fn(L)p Fr(,)h Fn(f)k Fm(2)c Fr(Hom)o(,)g(w)o(e)g(de\014ne)f Fn(cf)5 b Fr(\()p Fn(x)p Fr(\))21 b(to)e(b)q(e)h(equal)f(to)0 540 y Fn(f)5 b Fr(\()p Fn(cx)p Fr(\).)18 b(No)o(w)e(dim)345 547 y Fk(L)384 540 y Fr(Hom)487 547 y Fk(K)526 540 y Fr(\()p Fn(L;)8 b(K)t Fr(\))14 b(=)g(1.)i(And)g(this)h(is)f(w)o (onderful!)0 674 y Fp(Lecture)k(31)c Fr(\(No)o(v)o(em)o(b)q(er)g(8,)g (1995\).)0 744 y(Scrib)q(e:)g(Nic)o(k)h(Mey)o(er)100 879 y(If)f Fn(K)i Fm(\032)c Fn(L)i Fr(is)g(\014nite)h(separable,)d (then)j(Hom)940 886 y Fk(K)978 879 y Fr(\()p Fn(L;)8 b(K)t Fr(\))17 b(is)f(spanned)f(b)o(y)i(T)l(r)1500 888 y Fk(L=K)1586 879 y Fr(.)100 950 y(Bac)o(k)g(to)g(function)g(\014elds)f (for)g(the)h(momen)o(t.)f(Let)i Fn(k)r(;)8 b(K)q(;)g(L)16 b Fr(b)q(e)i(as)e(ab)q(o)o(v)o(e.)g(T)l(ak)o(e)h Fn(\025)e Fm(2)f Fr(Di\013)1811 957 y Fk(K)1861 950 y Fm(\000)d Fr(0.)0 1020 y(T)l(o)18 b(eac)o(h)g Fn(\025)g Fr(is)g(asso)q(ciated)g (a)h(canonical)e(divisor)g Fn(W)1002 1027 y Fk(\025)1045 1020 y Fr(=)1101 982 y Fh(P)1153 1035 y Fk(P)5 b Fj(2)p
Fk(C)1239 1040 y Fc(K)1283 1020 y Fn(n)p Fr(\()p Fn(P)i Fr(\))12 b Fm(\001)h Fn(P)7 b Fr(,)18 b(where)f Fn(P)26 b Fr(ranges)17 b(o)o(v)o(er)0 1098 y(the)23 b(closed)f(p)q(oin)o(ts)f (of)i Fn(C)499 1105 y Fk(K)560 1098 y Fr(and)f Fn(n)p Fr(\()p Fn(P)7 b Fr(\))24 b(=)g(max)8 b Fn(n)13 b Fm(2)h Fp(Z)g Fm(j)g Fn(\025)g Fr(=)f(0)k(on)f Ff(m)1369 1076 y Fj(\000)p Fk(n)1369 1112 y(P)1441 1098 y Fm(\032)e Fp(A)1537 1105 y Fg(K)1575 1098 y Fr(.)22 b(No)o(w)h(deg)8 b Fn(W)1861 1105 y Fk(\025)1911 1098 y Fr(=)0 1169 y(2)p Fn(g)49 1176 y Fk(K)98 1169 y Fm(\000)j Fr(2[)p Fn(K)k Fm(\\)289 1155 y Fr(\026)288 1169 y Fn(k)g Fr(:)f Fn(k)r Fr(])f(=)465 1131 y Fh(P)526 1169 y Fn(n)p Fr(\()p Fn(P)7 b Fr(\))h(deg)h Fn(P)e Fr(.)100 1239 y(No)o(w)17 b(let's)h(get)h(bac)o (k)e(to)h(pro)o(ving)f(the)h(Hurwitz)g(F)l(orm)o(ula.)d(Our)h (strategy:)i(tak)o(e)h(a)e Fn(\025)g Fm(2)f Fr(Di\013)1898 1246 y Fk(K)1936 1239 y Fr(,)0 1309 y(use)g(it)h(to)g(conco)q(ct)g(a)g Fn(\026)c Fm(2)h Fr(Di\013)591 1316 y Fk(L)621 1309 y Fr(,)j(and)f(use)g(it)g(to)h(calculate)g Fn(g)1177 1316 y Fk(L)1206 1309 y Fr(.)100 1380 y(T)l(o)24 b(this)g(end,)g(c)o(ho)q (ose)g Fn(t)k Fr(:)f Fn(L)g Fm(!)g Fn(K)t Fr(.)e(It's)f(a)h(surjectiv)o (e)f Fn(K)t Fr(-linear)f(map)h(\(i.e.,)h Fn(t)g Fr(is)f(unique)0 1449 y(up)e(to)h Fn(L)179 1431 y Fj(\003)201 1449 y Fr(\).)g(W)l(rite)g Fn(t)420 1431 y Fj(0)457 1449 y Fr(=)h Fn(t)15 b Fm(\012)f Fr(1)24 b(:)g Fp(A)736 1456 y Fg(L)789 1436 y Fm(\030)789 1452 y Fr(=)851 1449 y Fn(L)15 b Fm(\012)939 1456 y Fk(K)993 1449 y Fp(A)1036 1456 y Fg(K)1097 1449 y Fm(!)23 b Fp(A)1213 1456 y Fg(K)1251 1449 y Fr(.)g(Sev)o(eral)e(things)h(here)g(should)f(b) q(e)0 1519 y(noted:)j(\014rst,)g(surjectivit)o(y)g(is)g(guaran)o(teed)f (since)h Fm(\012)h Fr(is)f(righ)o(t-exact\);)g(second,)f Fn(t)1620 1501 y Fj(0)1659 1519 y Fr(is)h Fp(A)1760 1526 y Fg(K)1798 1519 y Fr(-linear.)0 1589 y(Finally)l(,)15 b(the)i(isomorphism)d(b)q(et)o(w)o(een)i Fp(A)792 1596 y Fg(L)839 1589 y Fr(and)g Fn(L)11 b Fm(\012)1020 1596 y Fk(K)1070 1589 y Fp(A)1113 1596 y Fg(K)1167 1589 y Fr(is)17 b(a)f(top)q(ological)h(isomorphism)c(as)k(w)o(ell)0 1659 y(-)f Fp(A)76 1666 y Fg(K)114 1637 y Fk(n)155 1645 y Fm(\030)155 1661 y Fr(=)208 1659 y Fm(\010)247 1640 y Fk(n)247 1671 y(i)p Fl(=1)313 1659 y Fn(!)344 1666 y Fk(i)372 1659 y Fm(\012)11 b Fp(A)465 1666 y Fg(K)503 1659 y Fr(.)100 1729 y(No)o(w)16 b(let)h Fn(\026)d Fr(=)f Fn(\025)f Fm(\016)f Fn(t)476 1711 y Fj(0)503 1729 y Fr(:)j Fp(A)574 1736 y Fg(L)618 1729 y Fm(!)f Fn(k)r Fr(.)100 1800 y Fp(Claim)p Fr(:)k Fn(\026)d Fm(2)g Fr(Di\013)451 1807 y Fk(L)481 1800 y Fr(.)100 1870 y Fp(Pro)r(of)p Fr(:)j(Con)o(tin)o(uit)o(y)h(follo)o(ws)h(form)g(the)g(top)q(ological)h (isomorphism)c(ab)q(o)o(v)o(e;)j Fn(\026)h Fr(is)f Fn(k)r Fr(-linear)e(b)o(y)0 1940 y(comp)q(osition;)e Fn(\026)p Fr(\()p Fn(L)p Fr(\))g(=)e(0)k(since)f Fn(t)643 1922 y Fj(0)657 1940 y Fr(\()p Fn(L)p Fr(\))e(=)g Fn(K)t Fr(.)i(So)g(w)o (e're)g(done.)100 2011 y Fp(Basic)k(F)-5 b(act)p Fr(:)17 b(W)l(rite)g Fp(D)572 2018 y Fk(t)604 2011 y Fr(=)658 1973 y Fh(P)710 2026 y Fk(Q)p Fj(2)p Fk(C)797 2031 y Fc(L)834 2011 y Fn(m)p Fr(\()p Fn(Q)p Fr(\))12 b Fm(\001)f Fn(Q)p Fr(,)18 b(where)e Fn(Q)h Fr(ranges)f(o)o(v)o(er)g(the)h(closed)g (p)q(oin)o(ts)f(of)0 2095 y Fn(C)36 2102 y Fk(L)65 2095 y Fr(,)h Fn(m)p Fr(\()p Fn(Q)p Fr(\))e(=)e(max)8 b Fn(n)14 b Fm(2)g Fp(Z)g Fm(j)f Fn(t)570 2077 y Fj(0)584 2095 y Fr(\()610 2093 y(^)603 2095 y Ff(m)641 2073 y Fj(\000)p Fk(n)641 2109 y(Q)700 2095 y Fr(\))h Fm(\032)805 2082
y Fr(^)786 2095 y Fm(O)826 2102 y Fk(p)865 2095 y Fr(\()p Fm(O)924 2102 y Fk(p)961 2095 y Fr(=)g Fn(K)h Fm(\\)c(O)1155 2102 y Fk(q)1177 2095 y Fr(\).)16 b(Then:)100 2169 y(i\))g(This)g(is)g (a)h(divisor)e(of)i Fn(L)p Fr(.)100 2240 y(ii\))f Fp(D)207 2247 y Fk(f)t Fj(\001)p Fk(t)274 2240 y Fr(=)d Fp(D)370 2247 y Fk(t)399 2240 y Fr(+)e(\()p Fn(f)5 b Fr(\),)18 b(where)e Fn(f)j Fm(2)14 b Fn(L)816 2222 y Fj(\003)839 2240 y Fr(.)100 2311 y(iii\))k Fn(W)226 2318 y Fk(\026)269 2311 y Fr(=)e Fn(i)p Fr(\()p Fn(W)407 2318 y Fk(\025)433 2311 y Fr(\))d(+)f Fp(D)560 2318 y Fk(t)577 2311 y Fr(,)18 b(where)f Fn(i)i Fr(is)e(the)h(group)f(homomorphism)d Fn(i)j Fr(:)f(Div)1566 2318 y Fk(K)1621 2311 y Fm(!)g Fr(Div)1765 2318 y Fk(L)1813 2311 y Fr(taking)0 2380 y Fn(P)21 b Fm(!)116 2343 y Fh(P)169 2395 y Fk(Q)p Fj(!)p Fk(P)282 2380 y Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\))p Fn(Q)p Fr(.)100 2455 y(\(Remark:)20 b(This)g(part)g(of)i(the)f(theorem)f(is)g (analogous)g(to)h(a)g(result)f(from)g(algebraic)g(n)o(um)o(b)q(er)0 2525 y(theory)l(.)d(If)h Fn(K)h Fm(\032)c Fn(L)g Fm(\032)g Fn(M)5 b Fr(,)18 b Fp(D)565 2534 y Fk(M)r(=K)679 2525 y Fr(=)d Fp(D)777 2534 y Fk(M)r(=L)878 2525 y Fm(\001)c Fp(D)947 2534 y Fk(L=K)1033 2525 y Fr(.)18 b(The)f(t)o(w)o(o)g(on)g (the)h(righ)o(t-hand)c(side)j(don't)g(sit)0 2595 y(in)f(the)h(same)f (ring)f(-)h(w)o(e)h(let)g Fn(i)p Fr(\()p Fp(D)630 2604 y Fk(L=K)716 2595 y Fr(\))e(=)e Fp(D)846 2604 y Fk(L=K)943 2595 y Fm(\001)e(O)1008 2602 y Fk(M)1052 2595 y Fr(\).)100 2670 y(iv\))17 b(2)p Fn(g)225 2677 y Fk(L)265 2670 y Fm(\000)11 b Fr(2[)p Fn(L)g Fm(\\)444 2657 y Fr(\026)443 2670 y Fn(k)k Fr(:)f Fn(k)r Fr(])f(=)h(\(2)p Fn(g)688 2677 y Fk(K)737 2670 y Fm(\000)d Fr(2[)p Fn(K)k Fm(\\)928 2657 y Fr(\026)927 2670 y Fn(k)g Fr(:)f Fn(k)r Fr(]\)[)p Fn(L)f Fr(:)h Fn(K)t Fr(])d(+)g(deg)d Fp(D)1394 2677 y Fk(t)1412 2670 y Fr(.)950 2790 y(64)p eop %%Page: 65 65 65 64 bop 100 50 a Fp(Pro)r(of)p Fr(:)11 b Fn(t)284 32 y Fj(0)312 50 y Fr(tak)o(es)j Fp(A)480 57 y Fg(L)524 50 y Fr(=)576 12 y Fh(Q)623 25 y Fj(0)623 65 y Fk(P)665 12 y Fh(Q)712 65 y Fk(Q)p Fj(!)p Fk(P)824 50 y Fn(L)858 57 y Fk(Q)906 50 y Fm(!)g Fp(A)1013 57 y Fg(K)1064 50 y Fr(=)1117 12 y Fh(Q)1164 25 y Fj(0)1186 50 y Fn(K)1228 57 y Fk(P)1262 50 y Fr(.)f(So)h Fn(t)1374 32 y Fj(0)1401 50 y Fr(is)g(really)f(made)g(up)g(of)h(maps)0 120 y Fn(t)18 127 y Fk(Q)66 120 y Fr(:)f Fn(L)127 127 y Fk(Q)175 120 y Fm(!)h Fn(K)281 127 y Fk(P)314 120 y Fr(,)j(where)f(eac)o(h)g Fn(t)619 127 y Fk(Q)669 120 y Fr(is)g Fn(K)761 127 y Fk(P)795 120 y Fr(-linear.)100 190 y(No)o(w,)g(w)o(e)g(pro)o(v)o(e)g (\(i\))h(and)f(\(iii\):)g(Let)i Fn(r)d Fm(2)f Fp(Z)p Fr(.)74 315 y Fn(r)h Fm(\024)f Fr(ord)236 322 y Fk(Q)270 315 y Fn(W)317 322 y Fk(\026)357 315 y Fm($)g Fn(\026)p Fr(\()477 313 y(^)470 315 y Ff(m)508 293 y Fj(\000)p Fk(r)508 329 y(Q)562 315 y Fr(\))g(=)g(0)357 409 y Fm($)g Fn(\025)p Fr(\()p Fn(t)487 416 y Fk(Q)521 409 y Fr(\()547 407 y(^)540 409 y Ff(m)578 387 y Fj(\000)p Fk(r)578 423 y(Q)632 409 y Fr(\)\))h(=)e(0)357 517 y Fm($)h Fn(t)439 524 y Fk(Q)473 517 y Fr(\()499 515 y(^)492 517 y Ff(m)530 496 y Fj(\000)p Fk(r)530 532 y(Q)584 517 y Fr(\))g Fm(\022)676 515 y Fr(^)669 517 y Ff(m)707 495 y Fj(\000)p Fr(ord)811 500 y Fc(P)839 495 y Fk(W)876 500 y Fc(\025)707 532 y Fk(P)357 625 y Fm($)g Fn(t)439 632 y Fk(Q)473 625 y Fr(\()499 623 y(^)492 625 y Ff(m)530 603 y Fj(\000)p Fr(ord)634
608 y Fc(P)662 603 y Fk(W)699 608 y Fc(\025)530 640 y Fk(P)726 625 y Fr(\))g Fm(\022)830 613 y Fr(^)811 625 y Fm(O)851 632 y Fk(p)357 725 y Fm($)g(\000)p Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\)ord)697 732 y Fk(P)730 725 y Fn(W)777 732 y Fk(\025)814 725 y Fr(+)k Fn(r)16 b Fm(\024)d Fr(ord)1027 732 y Fk(Q)1060 725 y Fp(D)1104 732 y Fk(t)1138 725 y Fr(\(since)1287 723 y(^)1280 725 y Ff(m)1318 732 y Fk(P)1362 725 y Fm(\001)1406 712 y Fr(^)1387 725 y Fm(O)1427 732 y Fk(q)1463 725 y Fr(=)1522 723 y(^)1515 725 y Ff(m)1553 699 y Fk(e)p Fl(\()p Fk(Q=P)5 b Fl(\))1553 739 y Fk(Q)1705 725 y Fr(b)o(y)16 b(def)t(\).)357 818 y Fm($)e Fn(r)i Fm(\024)d Fr(ord)583 825 y Fk(Q)617 818 y Fp(D)661 825 y Fk(t)690 818 y Fr(+)e Fn(e)p Fr(\()p Fn(Q=P)c Fr(\))12 b Fm(\001)f Fr(ord)1013 825 y Fk(P)1046 818 y Fn(W)1093 825 y Fk(\025)0 942 y Fr(Therefore,)j(for)h(eac)o(h)g Fn(Q)p Fr(,)h(ord)565 949 y Fk(Q)607 942 y Fn(W)654 949 y Fk(\026)694 942 y Fr(=)e(ord)819 949 y Fk(Q)861 942 y Fp(D)905 949 y Fk(t)931 942 y Fr(+)8 b Fn(e)p Fr(\()p Fn(Q=P)f Fr(\))j Fm(\001)e Fr(ord)1246 949 y Fk(P)1288 942 y Fn(W)1335 949 y Fk(\025)1361 942 y Fr(.)15 b(Th)o(us)f Fp(D)1559 949 y Fk(t)1590 942 y Fr(=)g Fn(W)1690 949 y Fk(\026)1725 942 y Fm(\000)8 b Fn(i)p Fr(\()p Fn(W)1855 949 y Fk(\025)1882 942 y Fr(\))16 b(is)0 1012 y(a)h(divisor.)e(So)h(w)o (e'v)o(e)g(pro)o(v)o(en)f(\(i\))i(and)f(\(iii\).)h(\(ii\))g(is)f(left)h (as)f(an)g(exercise)h(for)f(the)h(reader.)100 1082 y(T)l(o)12 b(get)i(\(iv\),)g(w)o(e)e(tak)o(e)i(degrees)e(on)h(b)q(oth)g(sides)f (of)h(\(iii\);)g(it)g(su\016ces)f(to)h(v)o(erify)g(that)h([)p Fn(L)f Fr(:)h Fn(K)t Fr(])8 b(deg)14 b(=)0 1151 y(deg)9 b Fm(\016)p Fn(i)16 b Fr(as)h(maps)e(from)h(Div)530 1158 y Fk(K)582 1151 y Fm(!)e Fp(Z)p Fr(.)i(W)l(ell,)459 1282 y(deg\()p Fn(i)p Fr(\()p Fn(P)7 b Fr(\)\))16 b(=)749 1235 y Fh(X)734 1342 y Fk(Q)p Fj(!)p Fk(P)844 1282 y Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\))h(deg)i Fn(Q)682 1436 y Fr(=)749 1388 y Fh(X)734 1495 y Fk(Q)p Fj(!)p Fk(P)844 1436 y Fn(e)p Fr(\()p Fn(Q=P)d Fr(\)[)p Fn(k)r Fr(\()p Fn(Q)p Fr(\))16 b(:)d Fn(k)r Fr(])682 1589 y(=)749 1542 y Fh(X)734 1649 y Fk(Q)p Fj(!)p Fk(P)844 1589 y Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\)[)p Fn(k)r Fr(\()p Fn(Q)p Fr(\))16 b(:)d Fn(k)r Fr(\()p Fn(P)7 b Fr(\)][)p Fn(k)r Fr(\()p Fn(P)g Fr(\))14 b(:)g Fn(k)r Fr(])682 1743 y(=)749 1695 y Fh(X)734 1802 y Fk(Q)p Fj(!)p Fk(P)844 1743 y Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\))p Fn(f)e Fr(\()p Fn(Q=P)i Fr(\))h(deg)t(\()p Fn(P)f Fr(\))682 1886 y(=)13 b(deg)q(\()p Fn(P)7 b Fr(\)[)p Fn(L)14 b Fr(:)f Fn(K)t Fr(])0 2010 y Fp(Lecture)20 b(32)c Fr(\(No)o(v)o(em)o(b)q(er)g(10,)g(1995\).)0 2080 y(Scrib)q(e:)g(Matt)h (Bak)o(er)100 2207 y(Here)g(is)g(a)g(sligh)o(tly)g(more)f(general)h (setup)f(to)i(consider)e([think)h(of)g(it)h(as)f(a)g(mo)q(di\014cation) f(of)i(our)0 2277 y(previous)d(assumptions].)100 2366 y(Let)j Fn(k)g Fm(\032)e Fn(K)t Fr(,)i(with)g Fn(K)j Fr(a)d(\014nitely)g(generated)g(extension)f(of)i Fn(k)g Fr(of)f(transcendence)f(degree)h(1,)f(as)0 2436 y(b)q(efore.)f(Let)i Fn(l)f Fr(and)f Fn(L)g Fr(b)q(e)h(\014nite)g(extensions)f(of)g Fn(k)i Fr(and)e Fn(K)t Fr(,)h(resp)q(ectiv)o(ely)l(.)100 2546 y(Then)f(the)g(form)o(ula)f(from)h(last)g(time:)314 2670 y(2)p Fn(g)363 2677 y Fk(L)404 2670 y Fm(\000)11 b Fr(2[)p Fn(L)f Fm(\\)p 582 2629 28 2 v 12 w Fn(k)15 b Fr(:)f Fn(k)r Fr(])f(=)h(\(2)p Fn(g)827 2677 y Fk(K)876 2670 y Fm(\000)d Fr(2[)p Fn(K)k Fm(\\)p 1066 2629 V 11 w Fn(k)g Fr(:)f Fn(k)r Fr(]\))d Fm(\003)g Fr([)p Fn(L)i
Fr(:)h Fn(K)t Fr(])d(+)f Fn(deg)r Fr(\()p Fm(D)1584 2677 y Fk(t)1602 2670 y Fr(\))p Fn(;)950 2790 y Fr(65)p eop %%Page: 66 66 66 65 bop 0 50 a Fr(where)15 b Fn(t)f Fr(:)g Fn(L)f Fm(!)h Fn(K)20 b Fr(is)15 b(a)h(K-linear)e(map,)g(can)i(b)q(e)g(soup)q(ed-up)e (to)i(our)f(new)g(setup)h(in)f(an)g(ob)o(vious)g(w)o(a)o(y)0 120 y(to)i(yield)f(the)h(follo)o(wing)e(form)o(ula:)203 206 y(2)p Fn(g)252 215 y Fk(L=l)326 206 y Fm(\000)10 b Fr(2[)p Fn(L)h Fm(\\)p 503 165 16 2 v 11 w Fn(l)k Fr(:)f Fn(l)q Fr(])f(=)h(\(2)p Fn(g)725 215 y Fk(K=k)815 206 y Fm(\000)c Fr(2[)p Fn(K)15 b Fm(\\)p 1004 165 28 2 v 11 w Fn(k)h Fr(:)d Fn(k)r Fr(]\))e Fm(\003)g Fr([)p Fn(L)j Fr(:)f Fn(K)t Fr(])e(+)g Fn(deg)1464 213 y Fk(l)1479 206 y Fr(\()p Fm(D)1536 213 y Fk(t)1554 206 y Fr(\))h Fm(\003)f Fr([)p Fn(l)j Fr(:)g Fn(k)r Fr(])p Fn(:)0 292 y Fr(The)k(Riemann-Hurwitz)e(form)o(ula)h(will)h(essen)o(tially)f(b)q (e)i(a)f(translation)f(of)i(this)f(form)o(ula)e(in)o(to)i(more)0 362 y(familiar)d(language)h(in)g(the)h(separable)e(case.)h(When)g Fn(k)i Fr(is)e(p)q(erfect,)h(w)o(e)g(ha)o(v)o(e)f(the)g(follo)o(wing)g (result.)100 471 y Fp(Theorem)p Fr(:)c(Supp)q(ose)h Fn(k)i Fr(is)f(p)q(erfect,)h(and)e(let)i Fn(L)1010 478 y Fk(sep)1084 471 y Fr(=)f Fm(f)p Fn(x)g Fm(2)g Fn(L)g Fr(:)f Fn(x)i Fr(is)f(separable)e(o)o(v)o(er)i Fn(K)t Fm(g)p Fr(.)g(Then)0 541 y Fn(g)24 550 y Fk(L=k)110 541 y Fr(=)f Fn(g)186 550 y Fk(L)213 555 y Fc(sep)268 550 y Fk(=k)313 541 y Fr(.)100 647 y Fp(Pro)r(of)p Fr(:)k(General)h(\014eld)g(theory)h(tells) g(us)f(that)i Fn(L)1048 629 y Fk(q)1087 647 y Fm(\032)e Fn(L)1178 654 y Fk(sep)1239 647 y Fr(,)h(where)f Fn(q)i Fr(=)e([)p Fn(L)f Fr(:)h Fn(L)1648 654 y Fk(sep)1708 647 y Fr(].)h(A)g(degree)0 717 y(calculation)d(similar)f(to)j(a)f (previous)e(calculation)i(done)f(in)g(class)h(sho)o(ws)e(that)i Fn(L)1551 724 y Fk(sep)1627 717 y Fr(=)d Fn(L)1714 699 y Fk(q)1736 717 y Fr(.)j(Th)o(us)e(w)o(e)0 787 y(ha)o(v)o(e)h Fn(g)139 796 y Fk(L=k)225 787 y Fr(=)d Fn(g)301 796 y Fk(L)328 786 y Fc(q)348 796 y Fk(=k)390 786 y Fc(q)427 787 y Fr(=)g Fn(g)503 796 y Fk(L)530 786 y Fc(q)550 796 y Fk(=k)609 787 y Fr(=)h Fn(g)686 796 y Fk(L)713 801 y Fc(sep)767 796 y Fk(=k)812 787 y Fr(.)100 896 y(Assume)h(no)o(w)h (that)h Fn(L)d Fm(\033)f Fn(K)21 b Fr(is)16 b Fp(separable)g Fr(and)g(tak)o(e)h Fn(t)d Fr(:=)f Fn(T)7 b(r)1320 905 y Fk(L=K)1407 896 y Fr(.)100 986 y Fp(Remark)p Fr(:)15 b(Separabilit)o(y)g(is)h(equiv)m(alen)o(t)g(to)h(the)g(trace)g(b)q (eing)f(nonzero)f(b)o(y)i(basic)e(\014eld)h(theory)l(.)100 1075 y(Recall)e(that)i Fm(D)393 1082 y Fk(t)425 1075 y Fr(=)477 1038 y Fh(P)538 1075 y Fn(m)p Fr(\()p Fn(Q)p Fr(\))c Fm(\001)f Fn(Q)p Fr(,)k(where)g(the)g(sum)f(is)h(tak)o(en)g(o)o (v)o(er)f(the)h(closed)g(p)q(oin)o(ts)f Fn(Q)h Fr(of)h Fn(C)1907 1082 y Fk(L)1936 1075 y Fr(,)0 1148 y(and)h(where)h Fn(m)p Fr(\()p Fn(Q)p Fr(\))h(is)e(de\014ned)g(as)h(the)g(largest)f Fn(n)f Fm(2)h Fp(Z)h Fr(suc)o(h)f(that)h Fn(t)1310 1155 y Fk(Q)1344 1148 y Fr(\()p Ff(m)1401 1127 y Fj(\000)p Fk(n)1401 1163 y(Q)1460 1148 y Fr(\))e Fm(\032)1561 1135 y Fh(c)1550 1148 y Fn(O)1588 1155 y Fk(P)1621 1148 y Fr(.)i(Here)g Fn(P)25 b Fr(is)18 b(the)0 1217 y(p)q(oin)o(t)e(in)g Fn(C)224 1224 y Fk(K)279 1217 y Fr(lying)g(under)f Fn(Q)p Fr(,)i(and)f Fn(t)728 1224 y Fk(Q)778 1217 y Fr(is)g(the)h(\(nonzero\)) g(trace)f(map)g(from)g Fn(L)1526 1224 y Fk(Q)1576 1217 y Fr(to)h Fn(K)1679 1224 y Fk(P)1712 1217 y Fr(.)100 1326 y(W)l(e)c(ha)o(v)o(e)g(a)g(couple)g(of)g(useful)g(form)o(ulas)e
(for)i(traces,)g(whic)o(h)g(can)g(b)q(e)g(v)o(eri\014ed)g(without)g(to) q(o)h(m)o(uc)o(h)0 1396 y(su\013ering:)100 1486 y(1.)i(If)h Fn(B)245 1468 y Fj(0)273 1486 y Fr(=)c Fn(B)374 1449 y Fh(N)429 1501 y Fk(A)470 1486 y Fn(A)507 1468 y Fj(0)521 1486 y Fr(,)k(then)f Fn(T)7 b(r)723 1495 y Fk(B)755 1485 y Fi(0)769 1495 y Fk(=)n(A)817 1485 y Fi(0)847 1486 y Fr(=)14 b Fn(T)7 b(r)958 1495 y Fk(B)r(=)n(A)1049 1449 y Fh(N)1113 1486 y Fn(id)1156 1493 y Fk(A)1186 1483 y Fi(0)1202 1486 y Fr(.)100 1571 y(2.)16 b(If)h Fn(B)f Fr(=)d Fn(B)349 1578 y Fl(1)380 1534 y Fh(L)444 1571 y Fn(B)482 1578 y Fl(2)504 1571 y Fr(,)j(then)h Fn(T)7 b(r)706 1580 y Fk(B)r(=)n(A)802 1571 y Fr(=)14 b Fn(T)7 b(r)913 1580 y Fk(B)943 1585 y Fd(1)963 1580 y Fk(=)n(A)1025 1571 y Fr(+)k Fn(T)c(r)1133 1580 y Fk(B)1163 1585 y Fd(2)1183 1580 y Fk(=)n(A)1233 1571 y Fr(.)100 1676 y(Also,)16 b(recall)g(that)h Fn(t)485 1658 y Fj(0)512 1676 y Fr(:)d Fp(A)583 1683 y Fk(L)627 1676 y Fm(!)g Fp(A)734 1683 y Fk(K)789 1676 y Fr(de\014ned)h(b)o(y)i Fn(t)1048 1658 y Fj(0)1075 1676 y Fr(=)d Fn(t)1154 1639 y Fh(N)1218 1676 y Fn(id)1261 1683 y Fg(A)1295 1688 y Fc(K)1346 1676 y Fr(w)o(as)i(built)g(up)g(from)g(the)g(maps)0 1746 y Fn(t)18 1753 y Fk(Q)66 1746 y Fr(:)d Fn(L)127 1753 y Fk(Q)175 1746 y Fm(!)h Fn(K)281 1753 y Fk(P)314 1746 y Fr(.)j(It)g(follo)o(ws)e(from)h(form)o(ula)f(1)h(that)h Fn(t)1030 1728 y Fj(0)1058 1746 y Fr(=)c Fn(T)7 b(r)1168 1755 y Fg(A)1202 1760 y Fc(L)1229 1755 y Fk(=)p Fg(A)1283 1760 y Fc(K)1318 1746 y Fr(.)100 1855 y Fp(Prop)p Fr(:)15 b(If)i Fn(Q)g Fr(lies)f(o)o(v)o(er)g Fn(P)23 b Fr(then)17 b Fn(L)d Fm(\033)f Fn(K)21 b Fr(is)16 b(inseparable)e(i\013)j Fn(L)1288 1862 y Fk(Q)1335 1855 y Fm(\033)d Fn(K)1430 1862 y Fk(P)1480 1855 y Fr(is)i(inseparable.)100 1965 y Fp(Pro)r(of)p Fr(:)g(Supp)q(ose)h Fn(L)g Fm(\033)f Fn(K)22 b Fr(is)c(inseparable.)e(Then)i Fn(T)7 b(r)1156 1974 y Fl(\()1174 1965 y Fn(L=K)t Fr(\))1298 1972 y Fl(=)1332 1965 y Fr(0,)18 b(so)g Fn(T)7 b(r)1510 1974 y Fl(\()1528 1965 y Fn(L)1562 1972 y Fk(Q)1596 1965 y Fn(=K)1663 1972 y Fk(P)1696 1965 y Fr(\))17 b(=)g(0.)h(Con-)0 2035 y(v)o(ersely)l(,)h (if)h Fn(L)f Fm(\033)g Fn(K)k Fr(is)d(separable,)e(then)h Fn(L)843 2042 y Fk(Q)896 2035 y Fr(=)g Fn(L)13 b Fm(\001)g Fn(K)1070 2042 y Fk(P)1123 2035 y Fm(\033)19 b Fn(K)1223 2042 y Fk(P)1276 2035 y Fr(is)g(separable)g(b)o(y)g(\014eld)g(theory)l (.)g(In)0 2105 y(particular,)c Fn(t)262 2112 y Fk(Q)309 2105 y Fr(=)f Fn(T)7 b(r)420 2114 y Fl(\()439 2105 y Fn(L)473 2112 y Fk(Q)506 2105 y Fn(=K)573 2112 y Fk(P)607 2105 y Fr(\))14 b Fm(6)p Fr(=)g(0.)100 2214 y(W)l(e)i(will)f(no)o(w)g (mak)o(e)h(some)f(remarks)f(that)i(apply)g(more)f(generally)l(,)g(for)g (example)h(to)g(the)g(situa-)0 2284 y(tion)h(of)g(algebraic)e(n)o(um)o (b)q(er)g(theory)l(.)h(So)h(let)g Fn(A)g Fr(b)q(e)g(a)g(complete)g(D)o (VR)f([in)h(our)f(case,)1654 2271 y Fh(c)1643 2284 y Fn(O)1681 2291 y Fk(P)1714 2284 y Fr(])h(with)g(\014eld)0 2354 y(of)k(fractions)f Fn(E)j Fr([whic)o(h)c(for)i(us)f(is)g Fn(K)734 2361 y Fk(P)767 2354 y Fr(],)h(let)g Fn(F)27 b Fm(\033)21 b Fn(E)i Fr(b)q(e)e(a)g(\014nite)f(separable)f(extension)h ([=)p Fn(L)1854 2361 y Fk(Q)1908 2354 y Fr(in)0 2428 y(disguise],)15 b(and)h(let)g Fn(B)j Fr([i.e.)544 2415 y Fh(c)533 2428 y Fn(O)571 2435 y Fk(Q)605 2428 y Fr(])e(b)q(e)g(the)f (unique)g(v)m(aluation)g(ring)g(of)h Fn(F)23 b Fr(lying)16 b(o)o(v)o(er)g Fn(A)p Fr(.)100 2538 y(Let)h Fn(e)d Fr(=)g Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))p Fn(;)8 b(f)22 b
Fr(=)14 b Fn(f)5 b Fr(\()p Fn(B)r(=)m(A)p Fr(\),)19 b(and)d(de\014ne)g Fn(m)p Fr([=)d Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\)])18 b(to)f(b)q(e)g(the)g(largest)f Fn(n)h Fr(suc)o(h)f(that)0 2603 y Fn(T)7 b(r)58 2612 y Fk(F)s(=E)141 2603 y Fr(\()p Ff(m)198 2582 y Fj(\000)p Fk(n)198 2618 y(B)257 2603 y Fr(\))22 b Fm(\032)f Fn(A)p Fr(.)h(This)f Fn(m)g Fr(is)f(called)h (the)h(`di\013eren)o(tial)e(exp)q(onen)o(t'.)h(Note)h(that)f Fn(m)h Fm(\025)f Fr(0,)h(since)0 2668 y Fn(T)7 b(r)q Fr(\()p Fn(B)r Fr(\))16 b Fm(\032)d Fn(A)p Fr(.)k(Our)f(old)g(quan)o (tit)o(y)g Fn(deg)731 2675 y Fk(l)746 2668 y Fr(\()p Fm(D)803 2675 y Fk(t)821 2668 y Fr(\))h(is)g(just)1006 2631 y Fh(P)1059 2683 y Fk(Q)1101 2668 y Fn(m)p Fr(\()p Fn(Q=P)7 b Fr(\))12 b Fm(\001)e Fn(deg)1395 2675 y Fk(l)1411 2668 y Fn(Q)p Fr(.)950 2790 y(66)p eop %%Page: 67 67 67 66 bop 100 52 a Fp(Remark)p Fr(:)17 b(In)h(n)o(um)o(b)q(er)f(theory) l(,)h Fm(D)785 61 y Fk(B)r(=)n(A)886 52 y Fr(=)f Ff(m)980 26 y Fk(m)p Fl(\()p Fk(B)r(=)n(A)p Fl(\))980 67 y Fk(B)1148 52 y Fr(is)h(called)h(the)g(`di\013eren)o(t';)e(it)i(is)f(an)h(ideal)0 137 y(of)f Fn(B)r Fr(.)g(Similarly)l(,)d(the)j(quan)o(tit)o(y)g(\001) 682 146 y Fk(B)r(=)n(A)780 137 y Fr(=)d Ff(m)872 111 y Fk(m)p Fl(\()p Fk(B)r(=)n(A)p Fl(\))p Fj(\001)p Fk(f)t Fl(\()p Fk(B)r(=)n(A)p Fl(\))872 151 y Fk(A)1185 137 y Fr(is)j(called)f(the)h(`discriminan)n(t';)d(it)j(is)f(an)0 206 y(ideal)f(of)h Fn(A)p Fr(.)100 318 y Fp(Def)p Fr(:)f Fn(B)j Fr(is)d(called)g Fp(unrami\014ed)f Fr(o)o(v)o(er)h Fn(A)h Fr(if)f Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))g(=)e(1)i(and)g Fn(A=)p Ff(m)1424 325 y Fk(A)1471 318 y Fm(\032)d Fn(B)r(=)p Ff(m)1626 325 y Fk(B)1678 318 y Fr(is)j(separable.)100 429 y(W)l(e)j(ha)o(v)o(e)g([)p Fn(F)26 b Fr(:)19 b Fn(E)s Fr(])f(=)g Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))d Fm(\001)e Fn(f)5 b Fr(\()p Fn(B)r(=)m(A)p Fr(\))21 b(=)e Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))c Fm(\001)d Fn(f)1212 436 y Fl(ins)1264 429 y Fr(\()p Fn(B)r(=)m(A)p Fr(\))j Fm(\001)e Fn(f)1467 436 y Fl(sep)1525 429 y Fr(\()p Fn(B)r(=)m(A)p Fr(\),)21 b(and)e Fn(B)r(=)m(A)i Fr(is)0 499 y(unrami\014ed)12 b(i\013)i Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))6 b Fm(\001)g Fn(f)514 506 y Fl(ins)568 499 y Fr(\()p Fn(B)r(=)m(A)p Fr(\))15 b(=)f(1.)g(W)l(e)g(sa)o(y)g(that)g Fn(B)r(=)m(A)h Fr(is)f Fp(rami\014ed)f Fr(if)h(it)h(is)e(not)i(unrami\014ed.)100 610 y(Next)i(time,)g(w)o(e)f(will)g(pro)o(v)o(e)f(the)i(follo)o(wing)f (theorem.)100 722 y Fp(Theorem)p Fr(:)e Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))h(=)f(0)f Fm(,)h Fn(B)i Fm(\033)e Fn(A)i Fr(is)g(unrami\014ed.)e(More)h(generally)l(,)h(w)o(e)g(ha)o(v)o (e)f Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))g Fm(\025)0 792 y Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))h Fm(\000)e Fr(1,)21 b(with)h(equalit)o(y)f(i\013)h Fn(B)i Fr(o)o(v)o(er)c Fn(A)i Fr(is)f Fp(tamely)k(rami\014ed)p Fr(,)c(i.e.)g(c)o(har\()p Fn(A=)p Ff(m)1688 799 y Fk(A)1720 792 y Fr(\))h(do)q(es)f(not)0 861 y(divide)16 b Fn(e)11 b Fm(\001)g Fn(f)231 868 y Fl(insep)323 861 y Fr(.)0 1000 y Fp(Lecture)20 b(33)c Fr(\(No)o(v)o(em)o(b)q(er)g(13,)g(1995\).)h(Scrib)q(e:)e(Diane)i (Maclagan)100 1140 y(Let)i Fn(A)f Fr(b)q(e)h(a)f(complete)g(discrete)g (v)m(aluation)f(ring,)h(and)f Fn(E)k Fr(its)d(\014eld)g(of)g (fractions.)f(Let)i Fn(F)26 b Fr(b)q(e)18 b(a)0 1210 y(\014nite)g(separable)g(extension)g(of)h Fn(E)s Fr(.)f(F)l(rom)f (earlier)h(this)g(semester)g(w)o(e)g(kno)o(w)g(that)h(there)g(is)f (exactly)0 1280 y(one)e(v)m(aluation)h(ring,)e Fn(B)r Fr(,)i(of)f Fn(F)24 b Fr(lying)16 b(o)o(v)o(er)g Fn(A)p
Fr(.)h(No)o(w)703 1415 y([)p Fn(F)k Fr(:)13 b Fn(E)s Fr(])h(=)f Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))p Fn(f)5 b Fr(\()p Fn(B)r(=)m(A)q Fr(\))100 1552 y(W)l(e)16 b(are)h(in)o (terested)e(in)517 1687 y Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))g(=)e(max)p Fm(f)p Fn(n)h Fm(2)g Fp(Z)i Fr(s.t.)h(T)l(r)o(\()p Fp(m)1228 1665 y Fj(\000)p Fk(n)1228 1701 y(B)1285 1687 y Fr(\))e Fm(\022)e Fn(A)p Fm(g)0 1834 y Fr(Last)18 b(time)g(w)o(e)f (de\014ned)g(the)h(di\013eren)o(t,)f Fm(D)812 1843 y Fk(B)r(=)n(A)911 1834 y Fr(=)f Fp(m)1015 1808 y Fk(m)p Fl(\()p Fk(B)r(=)n(A)p Fl(\))1015 1848 y Fk(B)1162 1834 y Fr(,)i(whic)o(h)f(is)g(analogous)g(to)h(the)g(di\013eren)o(t)0 1903 y(from)e(Algebraic)g(Num)o(b)q(er)f(Theory)l(.)100 1975 y(In)f(Num)o(b)q(er)g(Theory)g(w)o(e)g(ha)o(v)o(e)g(the)h (situation)f(where)g Fn(L)g Fm(\022)f Fn(K)19 b Fr(is)14 b(a)h(\014nite)f(extension)g(of)h(n)o(um)o(b)q(er)0 2045 y(\014elds,)21 b(and)h Fn(B)j Fr(and)c Fn(A)i Fr(are)f(their)g(resp)q (ectiv)o(e)h(rings)e(of)h(in)o(tegers.)f(If)i Fn(E)i Fr(is)d(a)g(fractional)g Fn(B)j Fr(ideal,)0 2116 y(w)o(e)20 b(de\014ne)g Fn(E)265 2083 y Fi(0)301 2116 y Fr(=)h Fm(f)p Fn(x)g Fm(2)f Fn(Ls:t:)p Fr(T)l(r\()p Fn(xE)s Fr(\))h Fm(\022)f Fn(A)p Fm(g)p Fr(.)h(The)f(classical)g(de\014nition)f(of)i (the)g(di\013eren)o(t)f(is)g(that)0 2191 y Fm(D)38 2200 y Fk(B)r(=)n(A)135 2191 y Fr(=)13 b Fn(B)227 2157 y Fi(0)244 2191 y Fr(.)j(W)l(e)h(can)f(write)h Fm(D)617 2200 y Fk(B)r(=)n(A)713 2191 y Fr(=)d Fn(B)806 2157 y Fi(0)839 2191 y Fr(as)829 2295 y Fh(Y)849 2399 y Fg(q)901 2342 y Fp(q)931 2322 y Fk(m)p Fl(\()991 2313 y(^)982 2322 y Fk(B)1012 2327 y Fc(q)1032 2322 y Fk(=)1063 2313 y Fl(^)1052 2322 y Fk(A)1082 2327 y Fc(p)1103 2322 y Fl(\))100 2531 y Fr(W)l(e)e(sa)o(y)g Fn(B)i Fr(is)e(unrami\014ed)e(o)o(v)o(er)h Fn(A)i Fr(if)f Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))p Fn(f)983 2538 y Fl(ins)1037 2531 y Fr(\()p Fn(B)r(=)m(A)p Fr(\))j(=)f(1)e(where)g Fn(f)1443 2547 y Fr(ins)1519 2531 y(is)g(de\014ned)f(as)h(follo)o(ws:)0 2600 y(giv)o(en)i(an)o(y)g(\014eld)g(extension)g Fn(l)g Fm(\022)g Fn(k)r Fr(,)g(w)o(e)g(de\014ne)g Fn(l)910 2607 y Fl(sep)982 2600 y Fr(to)g(b)q(e)h(the)g(elemen)o(ts)e(of)i Fn(l)g Fr(separable)e(o)o(v)o(er)g Fn(k)r Fr(.)h(No)o(w)0 2670 y Fn(f)5 b Fr(\()p Fn(B)r(=)m(A)p Fr(\))16 b(=)e([)p Fn(l)g Fr(:)g Fn(k)r Fr(])i(where)g Fn(l)f Fr(=)e Fn(B)r(=)p Fp(m)704 2677 y Fk(B)754 2670 y Fr(and)j Fn(k)f Fr(=)f Fn(A=)p Fp(m)1056 2677 y Fk(A)1087 2670 y Fr(.)j Fn(f)1142 2686 y Fr(ins)1206 2670 y(\()p Fn(B)r(=)m(A)p Fr(\))e(=)f([)p Fn(l)g Fr(:)g Fn(l)1497 2677 y Fr(sep)1569 2670 y(].)950 2790 y(67)p eop %%Page: 68 68 68 67 bop 100 50 a Fn(B)22 b Fr(is)e(said)g(to)g(b)q(e)h(rami\014ed)e (if)h(it)h(is)f(not)g(unrami\014ed,)e(and)i Fn(B)j Fr(is)c(tamely)i (rami\014ed)e(o)o(v)o(er)g Fn(A)i Fr(if)0 120 y(c)o(har)o(\()p Fn(A=)p Fp(m)223 127 y Fk(A)255 120 y Fr(\))h(do)q(es)g(not)g(divide)f Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))p Fn(f)845 136 y Fr(ins)911 120 y(\()p Fn(B)r(=)m(A)p Fr(\).)i(If)f(the)g(residue)f(c)o (haracteristic)g(is)g(zero,)g(all)0 189 y(extensions)16 b(are)g(tamely)h(rami\014ed.)d(The)j(opp)q(osite)f(of)h(tamely)f (rami\014ed)f(is)h(wildly)g(rami\014ed.)100 259 y(F)l(or)e(more)h (details)g(on)g(this,)g(and)g(on)g(comp)q(ositums)f(of)h(of)h (\(un\)rami\014ed)e(extensions,)h(see)g(Lang)100 329 y(W)l(e)h(th)o(us)g(break)g(up)g(\014eld)g(extensions)g(as)g(follo)o (ws:)763 455 y Fn(E)g Fm(\022)e Fn(F)901 462 y Fk(ur)960 455 y Fm(\022)g Fn(F)1045 462 y Fk(tr)1096 455 y Fm(\022)f
Fn(F)0 581 y Fr(where)j Fn(F)176 588 y Fk(ur)238 581 y Fr(is)g(the)h(unrami\014ed)d(piece,)j(and)f Fn(F)891 588 y Fk(tr)944 581 y Fr(is)g(the)h(tamely)g(rami\014ed)e(piece.)100 651 y(It)h(is)g(imp)q(ortan)o(t)f(to)i(distinguish)d(b)q(et)o(w)o(een)i (the)h(c)o(haracteristic)e(of)h(the)h(residue)e(class)g(and)h(that)0 721 y(of)h(the)g(\014eld)e(extension.)0 790 y Fp(Theorem:)0 860 y Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))g(=)f(0)f Fm(,)h Fn(B)19 b Fr(is)d(unrami\014ed)e(o)o(v)o(er)i Fn(A)0 930 y(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))f Fm(\025)f Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))e Fm(\000)f Fr(1)0 1000 y Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))k(=)f Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))h Fm(,)f Fn(B)19 b Fr(is)d(tamely)h(rami\014ed)d (o)o(v)o(er)i Fn(A)p Fr(.)0 1069 y Fp(Pro)r(of:)100 1139 y Fr(T)l(ensor)f(T)l(r)313 1148 y Fk(B)r(=)n(A)410 1139 y Fr(:)e Fn(B)j Fm(!)e Fn(A)j Fr(with)f Fn(A=)p Fp(m)833 1146 y Fk(A)882 1139 y Fr(o)o(v)o(er)f Fn(A)p Fr(.)i(W)l(e)g(get)657 1272 y(T)l(r)708 1281 y Fk(B)r(=)n(A)801 1272 y Fm(\012)11 b Fn(id)j Fr(=)g(T)l(r)p 1012 1259 32 2 v 1012 1288 a Fk(B)r(=)p 1064 1259 30 2 v(A)1111 1272 y Fr(:)p 1138 1232 41 2 v 13 w Fn(B)i Fm(!)p 1256 1232 38 2 v 14 w Fn(A)0 1420 y Fr(where)p 147 1380 41 2 v 19 w Fn(B)k Fr(=)e Fn(B)r(=)p Fp(m)376 1427 y Fk(B)428 1420 y Fr(and)p 528 1380 38 2 v 19 w Fn(A)g Fr(=)g Fn(A=)p Fp(m)751 1427 y Fk(A)783 1420 y Fr(.)h(No)o(w)g Fp(m)981 1394 y Fk(e)p Fl(\()p Fk(B)r(=)n(A)p Fl(\))981 1435 y Fk(B)1130 1420 y Fr(=)f Fp(m)1236 1427 y Fk(A)1268 1420 y Fn(B)r Fr(,)h(so)g Fn(x)f Fm(2)h Fp(m)1552 1427 y Fk(B)1604 1420 y Fr(means)p 1759 1393 29 2 v 18 w Fn(x)g Fm(2)p 1857 1380 41 2 v 18 w Fn(B)j Fr(is)0 1490 y(nilp)q(oten)o(t.)753 1560 y(T)l(r)804 1569 y Fk(B)r(=)n(A)887 1560 y Fr(\()p Fp(m)955 1567 y Fk(B)988 1560 y Fr(\))14 b(=)p 1074 1522 25 2 v 14 w(0)g Fm(2)p 1160 1520 38 2 v 14 w Fn(A)774 1663 y Fr(T)l(r)826 1672 y Fk(B)r(=)n(A)908 1663 y Fr(\()p Fp(m)976 1670 y Fk(B)1010 1663 y Fr(\))g Fm(\022)f Fp(m)1144 1670 y Fk(A)745 1772 y Fp(m)794 1751 y Fj(\000)p Fl(1)794 1787 y Fk(A)846 1772 y Fr(T)l(r)898 1781 y Fk(B)r(=)n(A)980 1772 y Fr(\()p Fp(m)1048 1779 y Fk(B)1082 1772 y Fr(\))h Fm(\022)f Fn(A)745 1945 y Fr(T)l(r)797 1954 y Fk(B)r(=)n(A)879 1945 y Fr(\()p Fp(m)947 1924 y Fj(\000)p Fl(1)947 1960 y Fk(A)1000 1945 y Fp(m)1049 1952 y Fk(B)1082 1945 y Fr(\))h Fm(\022)f Fn(A)706 2063 y Fr(T)l(r)757 2072 y Fk(B)r(=)n(A)840 2063 y Fr(\()p Fp(m)908 2037 y Fj(\000)p Fk(e)p Fl(\()p Fk(B)r(=)n(A)p Fl(\)[phone] y Fk(B)1121 2063 y Fr(\))h Fm(\022)g Fn(A)0 2166 y Fr(So)i Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))f Fm(\025)f Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))f Fm(\000)d Fr(1)100 2235 y(W)l(e)16 b(no)o(w)g(consider)f(when)h(equalit)o(y)h(is)f(reac)o(hed.)f(If)i (this)f(is)g(not)h(the)g(case,)f(then)770 2362 y Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))f Fm(\025)e Fn(e)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))699 2497 y Fm(,)h Fr(T)l(r)814 2506 y Fk(B)r(=)n(A)897 2497 y Fr(\()p Fp(m)965 2471 y Fj(\000)p Fk(e)p Fl(\()p Fk(B)r(=)n(A)p Fl(\))965 2512 y Fk(B)1128 2497 y Fr(\))g Fm(\022)f Fn(A)763 2670 y Fm(,)h Fr(T)l(r)878 2679 y Fk(B)r(=)n(A)961 2670 y Fr(\()p Fn(B)r Fr(\))h Fm(\022)e Fp(m)1155 2677 y Fk(A)950 2790 y Fr(68)p eop %%Page: 69 69 69 68 bop 830 50 a Fm(,)13 b Fr(T)l(r)p 945 37 32 2 v 16 x Fk(B)q(=)p 996 37 30 2 v(A)1043 50 y Fr(=)h(0)690 224 y Fm(,)g Fn(p)p Fm(j)p Fn(e)p Fr(\()p Fn(B)r(=)m(A)p
Fr(\))k(or)e(T)l(r)p 1084 211 32 2 v 17 x Fk(B)q(=)p 1135 211 30 2 v(A)1182 224 y Fr(=)e(0)0 337 y Fm(,)g Fn(B)r(=)m(A)j Fr(is)f(tamely)h(rami\014ed.)100 406 y(The)11 b(form)o(ula)e(for)j Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))g(in)f(the)h (wild)f(case)g(dep)q(ends)f(on)h(higher)g(rami\014cation.)e(The)i (reference)0 476 y(for)16 b(this)g(material)g(is)g(Serre's)f Fq(Corps)k(L)m(o)m(c)m(aux)p Fr(,)g(whic)o(h)c(is)h(a)o(v)m(ailable)g (in)g(translation.)100 546 y(W)l(e)22 b(mak)o(e)g(the)h(assumption)d (that)j Fn(A=)p Fp(m)913 553 y Fk(A)967 546 y Fr(is)e(p)q(erfect.)i (This)f(assumption)e(is)i(also)g(made)f(b)o(y)0 615 y(Serre.)100 685 y(Finite)15 b(separable)f(extensions)g Fn(F)23 b Fr(of)16 b Fn(E)i Fr(corresp)q(ond)13 b(to)j(\014nite)f(sets)h Fn(S)i Fr(together)d(with)h(a)f(con)o(tin-)0 755 y(uous)e(transitiv)o (e)h Fn(G)p Fm(\000)p Fr(action)f(on)p Fn(S)s Fr(.)h(\(Recall)g(21st)g (cen)o(tury)f(Galois)h(Theory)f(earlier)h(in)f(the)i(semester\).)100 829 y(F)l(or)d Fn(\033)k Fm(2)e Fn(G)p Fr(,)g Fn(\033)i Fm(6)p Fr(=)e Fn(id)g Fr(let)g Fn(j)s Fr(\()p Fn(\033)r Fr(\))g(b)q(e)h(the)f(largest)f Fn(j)j Fr(for)e(whic)o(h)e Fn(\033)17 b Fr(induces)12 b(the)i(iden)o(tit)o(y)g(on)f Fn(A)1817 811 y Fj(0)1831 829 y Fn(=)p Fp(m)1905 805 y Fk(j)1905 844 y(A)1935 834 y Fi(0)0 899 y Fr(where)j Fn(A)181 881 y Fj(0)212 899 y Fr(is)g(the)h(ring)e(sitting)i(ab)q(o)o (v)o(e)f Fn(A)h Fr(in)f Fn(E)s Fr(.)g(Then)426 1054 y Fn(m)p Fr(\()p Fn(B)r(=)m(A)p Fr(\))p Fn(f)5 b Fr(\()p Fn(B)r(=)m(A)p Fr(\))17 b(=)898 1007 y Fh(X)843 1114 y Fk(\033)q Fj(2)p Fk(G;\033)q Fj(6)p Fl(=)p Fk(id)1035 1054 y Fr(+\(#)p Fn(S)c Fm(\000)e Fr(#)p Fn(s)1294 1034 y Fk(\033)1320 1054 y Fr(\))1386 1021 y Fn(j)s Fr(\()p Fn(\033)r Fr(\))p 1345 1043 173 2 v 1345 1089 a Fn(e)p Fr(\()p Fn(A)1424 1074 y Fj(0)1440 1089 y Fn(=)m(A)p Fr(\))100 1235 y(Plugging)k(this)h(in)o(to)g(the)h(Riemann-Huriwitz)d (form)o(ula,)h(w)o(e)h(get:)88 1391 y(2)p Fn(g)137 1400 y Fk(L=l)211 1391 y Fm(\000)11 b Fr(2[)p Fn(L)g Fm(\\)p 389 1350 16 2 v 11 w Fn(l)k Fr(:)e Fn(l)q Fr(])h(=)f(\(2)p Fn(g)610 1400 y Fk(K=k)700 1391 y Fm(\000)e Fr(2[)p Fn(K)k Fm(\\)p 890 1350 28 2 v 11 w Fn(k)g Fr(:)f Fn(k)r Fr(]\))1026 1357 y([)p Fn(L)f Fr(:)h Fn(K)t Fr(])p 1026 1379 149 2 v 1044 1425 a([)p Fn(l)g Fr(:)g Fn(k)r Fr(])1192 1391 y(+)1339 1343 y Fh(X)1241 1461 y Fr(rami\014ed)o Fk(Q)p Fj(2)p Fk(C)1508 1391 y Fr(\(deg)1602 1403 y Fk(l)1617 1391 y Fn(Q)p Fr(\))p Fn(m)p Fr(\()p Fn(Q=P)7 b Fr(\))100 1582 y(Often)23 b(this)g(form)o(ula)e(can)i(b)q(e)g(simpli\014ed)e(a)i (lot)g(-)g(for)g(example,)f(it)h(is)g(often)g(the)h(case)f(that)0 1652 y([)p Fn(L)15 b Fm(\\)p 111 1611 16 2 v 15 w Fn(l)25 b Fr(:)e Fn(l)q Fr(])h(=)f([)p Fn(K)c Fm(\\)p 427 1611 28 2 v 15 w Fn(k)26 b Fr(:)d Fn(k)r Fr(])h(=)f(1.)g(This)e(happ)q(ens)h (in)g(particular)f(when)p 1463 1611 V 22 w Fn(k)k Fr(=)f Fn(k)r Fr(.)e(W)l(e)g(then)h(ha)o(v)o(e)0 1721 y([)p Fn(l)c Fr(:)g Fn(k)r Fr(])e(=)i(1,)g(ho)o(w)o(ev)o(er)f(there)h(are)g (in)o(teresting)f(cases)h(where)g(this)g(is)g(not)g(the)h(case.)f(If)g (it)h(is)f(tamely)0 1791 y(rami\014ed,)c(w)o(e)h(can)g(replace)g Fn(m)p Fr(\()p Fn(Q=P)7 b Fr(\))18 b(b)o(y)e Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\))12 b Fm(\000)f Fr(1.)0 1920 y Fp(Lecture)20 b(34)c Fr(\(No)o(v)o(em)o(b)q(er)g(15,)g(1995\).)0 1990 y(Scrib)q(e:)g(Diane)g(Maclagan)100 2120 y(Reference)j(for)h(to)q(da)o (y:)f(Artin,)h Fo(Algebraic)f(Num)o(b)q(ers)f(and)h(Algebraic)g(F)l (unctions)p Fr(.)f(Artin)i(also)0 2189 y(considers)11 b(things)g(in)h(sligh)o(tly)f(more)h(generalit)o(y)l(.)f(W)l(e)i(will)e (only)i(co)o(v)o(er)e(the)i(imp)q(ortan)o(t)e(results,)g(rather)0
2259 y(than)16 b(giving)g(a)h(general)f(surv)o(ey)f(of)i(the)g (\014eld.)100 2329 y(W)l(e)f(consider)g(the)g(situation:)875 2381 y Fn(l)59 b Fm(\032)d Fn(L)866 2441 y Fm([)145 b([)869 2501 y Fn(k)54 b Fm(\032)c Fn(K)0 2600 y Fr(where)16 b([)p Fn(L)e Fr(:)f Fn(l)q(k)r Fr(])j(is)g(\014nite.)g(Artin)h(do)q(es) f(not)h(require)e(this)h(assumption.)100 2670 y(W)l(e)g(sa)o(y)g(that)h (w)o(e)g(ha)o(v)o(e)f(a)g(\\constan)o(t)g(\014eld)g(extension")g(if)h Fn(L)c Fr(=)h Fn(l)q(k)r Fr(.)950 2790 y(69)p eop %%Page: 70 70 70 69 bop 0 50 a Fp(Theorem.)27 b Fo(Let)13 b Fn(k)i Fm(\032)f Fn(K)i Fo(b)q(e)c(a)h(\014nitely)f(generated)f(\014eld)h (extension)g(of)h(transcendence)e(degree)h(1)g(and)0 120 y(let)k Fn(l)f Fm(\033)e Fn(k)18 b Fo(b)q(e)e(a)f(\014nite)h (separable)e(extension)i(for)f(whic)o(h)g Fn(L)e Fr(=)h Fn(l)c Fm(\012)1250 127 y Fk(k)1284 120 y Fn(K)20 b Fo(is)15 b(a)h(\014eld)f(\(e.g.)h(if)p 1697 79 28 2 v 15 w Fn(k)11 b Fm(\\)f Fn(K)18 b Fr(=)13 b Fn(k)r Fo(\).)0 189 y(Then)j Fn(g)154 198 y Fk(L=l)231 189 y Fr(=)d Fn(g)307 198 y Fk(K=k)386 189 y Fo(.)100 287 y Fq(Pr)m(o)m(of:)161 433 y Fr(\(2)p Fn(g)229 442 y Fk(L=l)303 433 y Fm(\000)e Fr(2[)p 392 393 16 2 v Fn(l)g Fm(\\)g Fn(L)j Fr(:)g Fn(l)q Fr(])f(=)h(\(2)p Fn(g)702 442 y Fk(K=k)792 433 y Fm(\000)d Fr(2[)p 881 393 28 2 v Fn(k)h Fm(\\)f Fn(K)18 b Fr(:)c Fn(k)r Fr(]\))1118 400 y([)p Fn(L)f Fr(:)h Fn(K)t Fr(])p 1118 422 149 2 v 1135 468 a([)p Fn(l)h Fr(:)f Fn(k)r Fr(])1283 433 y(+)1348 386 y Fh(X)1333 493 y Fk(Q)p Fj(7!)p Fk(P)1443 433 y Fn(m)p Fr(\()p Fn(Q=P)7 b Fr(\))h(deg)1713 446 y Fk(l)1736 433 y Fn(Q:)0 615 y Fr(No)o(w)19 b([)p 130 574 16 2 v Fn(l)13 b Fm(\\)g Fn(L)k Fr(:)h Fn(l)q Fr(])f(=)h([)p 405 574 28 2 v Fn(k)c Fm(\\)e Fn(K)22 b Fr(:)c Fn(k)r Fr(])g(\(exercise\).)h(So)g(w)o(e)f(need)h(to)g(sho)o (w)1340 577 y Fh(P)1392 630 y Fk(Q)p Fj(7!)p Fk(P)1505 615 y Fn(m)p Fr(\()p Fn(Q=P)7 b Fr(\))h(deg)1774 627 y Fk(l)1798 615 y Fn(Q)18 b Fr(=)f(0.)0 685 y(W)l(e)g(kno)o(w)f(it)h (is)f(non-negativ)o(e,)f(so)h Fn(g)708 694 y Fk(L=l)785 685 y Fm(\025)d Fn(g)861 694 y Fk(K=k)940 685 y Fr(.)j(W)l(e)h(sho)o(w) e(gen)o(us)h(can't)g(go)h(up.)454 816 y Fn(g)e Fr(=)e(dim)629 823 y Fk(k)661 816 y Fp(A)704 823 y Fk(k)729 816 y Fn(=)p Fr(\()p Fn(K)i Fr(+)894 803 y(^)880 816 y Fm(O)q Fr(\))493 910 y(=)e(dim)629 917 y Fk(l)652 910 y Fn(l)f Fm(\012)f Fr(\()p Fp(A)791 917 y Fk(k)816 910 y Fn(=K)k Fr(+)961 898 y(^)948 910 y Fm(O)q Fr(\))493 1006 y(=)e(dim)629 1013 y Fk(l)644 1006 y Fr(\(\()p Fn(l)f Fm(\012)748 1013 y Fk(k)784 1006 y Fp(A)827 1013 y Fk(k)852 1006 y Fr(\))p Fn(=)p Fr(\()p Fn(l)g Fm(\012)981 1013 y Fk(k)1016 1006 y Fn(K)915 1034 y Fh(|)p 937 1034 29 6 v 29 w({z)p 1010 1034 V 29 w(})975 1081 y Fk(L)1071 1006 y Fr(+)p Fn(l)f Fm(\012)1175 1013 y Fk(k)1215 992 y Fr(^)1211 1006 y Fn(\022)q Fr(\)\))493 1163 y(=)i(dim)629 1170 y Fk(l)652 1163 y Fp(A)695 1170 y Fk(L)725 1163 y Fn(=)p Fr(\()p Fn(L)e Fr(+)g(\()17 b(some)f(subspace)f(of)1304 1150 y(^)1290 1163 y Fm(O)j Fr(for)e Fn(L)p Fr(\)\))493 1253 y Fm(\025)d Fn(g)569 1262 y Fk(L=l)100 1350 y Fr(where)j(the)g(last)h (t)o(w)o(o)f(lines)g(follo)o(w)g(b)q(ecause)421 1471 y Fn(l)c Fm(\012)487 1478 y Fk(k)536 1459 y Fr(^)523 1471 y Fm(O)j(\022)e Fn(l)f Fm(\012)696 1478 y Fk(k)731 1471 y Fp(A)774 1478 y Fk(k)813 1471 y Fr(=)i Fn(l)e Fm(\012)932 1478 y Fk(k)967 1471 y Fn(K)j Fm(\012)1063 1478 y Fk(k)1098 1471 y Fn(A)1135 1478 y Fk(k)1174 1471
y Fr(=)e Fn(L)e Fm(\012)g Fp(A)1364 1478 y Fk(k)1403 1471 y Fr(=)i Fp(A)1498 1478 y Fk(L)616 1593 y Fr(image)o(\()p Fn(l)f Fm(\012)853 1580 y Fr(^)839 1593 y Fm(O)18 b Fr(in)e Fp(A)998 1600 y Fk(L)1028 1593 y Fr(\))e Fm(\022)g Fr(\()1147 1580 y(^)1133 1593 y Fm(O)k Fr(for)e Fn(L)p Fr(\))p Fn(:)100 1733 y Fr(Consider)f(the)h(follo)o(wing)g(situation:)100 1803 y Fn(L)d Fr(=)h Fn(l)q(k)r Fr(,)p 274 1763 28 2 v 16 w Fn(k)e Fm(\\)g Fn(K)17 b Fr(=)d Fn(k)r Fr(,)p 527 1763 16 2 v 16 w Fn(l)e Fm(\\)f Fn(L)j Fr(=)f Fn(l)q Fr(.)100 1873 y(Then:)100 1943 y Fn(g)124 1952 y Fk(L=l)200 1943 y Fm(\024)h Fn(g)277 1952 y Fk(K=k)369 1943 y Fm(\021)g Fn(g)446 1952 y Fk(L=l)508 1943 y Fn(mod)p Fr(\()p Fn(p)e Fm(\000)f Fr(1\))p Fn(=)p Fr(2)16 b(\(T)l(ate)i(1952\))100 2012 y Fn(g)124 2021 y Fk(L=l)200 2012 y Fr(=)c Fn(g)277 2021 y Fk(K=k)372 2012 y Fr(if)j Fn(k)h Fr(is)e(p)q(erfect)h(\(see)g (Artin\).)100 2082 y Fn(K)g Fm(\033)d Fn(k)k Fr(is)e(called)g(conserv)m (ativ)o(e)h(if)f(w)o(e)h(ha)o(v)o(e)e(equalit)o(y)i(for)f(all)g Fn(L)e Fm(\033)g Fn(l)q Fr(.)100 2151 y Fp(Exercise:)j Fn(K)k Fr(is)16 b(conserv)m(ativ)o(e)g(i\013)g Fn(g)817 2160 y Fk(K=k)910 2151 y Fr(=)d(0)k(or)f Fn(C)1101 2158 y Fk(K)1155 2151 y Fr(is)h(nonsingular.)100 2220 y(Recall)f(from)g (last)g(time)g Fn(m)p Fr(\()p Fn(Q=P)7 b Fr(\))15 b(=)e Fn(e)p Fr(\()p Fn(Q=P)7 b Fr(\))13 b Fm(\000)d Fr(1)17 b(\(tamely)g(rami\014ed\).)100 2290 y(If)c Fm(G)176 2297 y Fk(P)222 2290 y Fr(is)g(the)g(Galois)g(group)f(of)h Fn(K)738 2266 y Fk(sep)734 2305 y(P)812 2290 y Fr(o)o(v)o(er)g Fn(K)958 2297 y Fk(P)991 2290 y Fr(,)g(and)g Fn(S)i Fr(is)e(a)g Fm(G)1273 2297 y Fk(K)1325 2290 y Fr(set)g(\(where)g Fm(G)1589 2297 y Fk(K)1641 2290 y Fr(is)f(the)i(absolute)0 2360 y(Galois)i(group)f(\()p Fn(Gal)q Fr(\()p Fn(K)459 2342 y Fk(sep)521 2360 y Fn(=K)t Fr(\)\),)i(then)539 2499 y Fm(f)p Fn(Q)g Fr(o)o(v)o(er)e Fn(P)7 b Fm(g)14 b( )-8 b(!)13 b(f)p Fr(orbits)j(of)g Fn(S)j Fr(under)d Fm(G)1354 2506 y Fk(P)1386 2499 y Fm(g)100 2600 y Fr(This)11 b(material)g(can)h(b)q(e)g(found)f(in)g(a)h(text)h(on)f(Algebraic)f (Num)o(b)q(er)g(Theory)h(or)f(V)l(aluation)g(Theory)0 2670 y(or)16 b(Serre)g Fo(Lo)q(cal)h(Fields)p Fr(.)950 2790 y(70)p eop %%Page: 71 71 71 70 bop 100 50 a Fr(W)l(e)16 b(ha)o(v)o(e)g(the)h(follo)o(wing)e (situation,)h(where)g(T)l(r)d(:)h Fn(L)g Fm(7!)f Fn(K)667 170 y(L)e Fm(\012)751 177 y Fk(K)800 170 y Fr(Di\013)881 179 y Fk(K=k)1025 141 y(T)5 b(r)1009 170 y Fm(\000)-8 b(!)49 b Fr(Di\013)1220 179 y Fk(L=l)796 229 y Fm([)189 b Fr(\(2\))702 289 y(\(Di\013)803 298 y Fk(K=k)881 289 y Fr(\))900 296 y Fk(p)0 378 y Fr(F)l(rom)15 b(Hartshorne,)g(p.300,)h (w)o(e)g(ha)o(v)o(e:)g(\(1\))h Fn(L)11 b Fm(\012)917 385 y Fk(K)966 378 y Fr(\012)1002 387 y Fk(K=k)1095 378 y Fm(!)i Fr(\012)1194 387 y Fk(L=l)100 447 y Fr(In)j(T)l(ate's)g(pap)q (er)g(he)h(dev)o(elops)e(maps)g(sho)o(wn)h(in)g(the)g(follo)o(wing)g (diagram:)717 547 y(\012)753 556 y Fk(K=k)914 519 y(C)890 547 y Fm(\000)-9 b(!)81 b Fr(Di\013)1131 556 y Fk(K=k)762 607 y Fm(")327 b([)709 666 y Fr(\012)745 673 y Fk(\022)764 678 y Fc(p)787 666 y Fn(=k)901 638 y Fk(C)930 643 y Fc(P)890 666 y Fm(\000)-9 b(!)50 b Fr(\(Di\013)1120 675 y Fk(K=k)1199 666 y Fr(\))1218 673 y Fk(p)100 766 y Fr(If)15 b Fn(K)j Fm(\033)c Fn(k)j Fr(separably)d(generated)g(then)i(injectiv)o(e)g (i\013)f Fn(P)22 b Fr(is)15 b(nonsingular.)e Fn(C)18 b Fr(is)d(an)g(isomorphism)0 836 y(i\013)h Fn(K)i Fm(\033)c
Fn(k)k Fr(is)e(separably)f(generated.)h Fn(C)766 843 y Fk(P)815 836 y Fr(is)h(an)f(isomorphism)d(i\013)k Fn(P)23 b Fr(is)16 b(nonsingular.)0 905 y(The)g(follo)o(wing)g(are)g (conjectures:)100 975 y Fp(Guess)i(1.)e Fr(If)h Fn(K)h Fm(\033)13 b Fn(k)18 b Fr(is)e(not)h(separably)e(generated)h(then)h Fn(c)c Fr(=)h(0.)100 1045 y Fp(Guess)k(2.)e Fn(K)q(=k)i Fr(separably)d(generated,)h(then)h(dim)1101 1052 y Fk(k)1134 1045 y Fr(k)o(er)8 b Fn(C)1245 1052 y Fk(P)1291 1045 y Fr(=)14 b(dim)1427 1052 y Fk(k)1460 1045 y Fr(cok)o(er)o Fn(C)1609 1052 y Fk(P)1642 1045 y Fr(\()p Fn(<)g Fm(1)p Fr(\).)100 1115 y Fp(Guess)k(3.)e Fr(These)g(t)o(w)o(o)g(guesses)g (follo)o(w)g(b)o(y)g(comparing)f(maps)g(\(1\))i(and)f(\(2\).)100 1184 y(There)g(is)g(th)o(us)f(still)h(researc)o(h)f(left)j(to)e(do)h (at)f(this)h(lev)o(el.)100 1254 y(Other)f(areas)f(of)i(imp)q(ortance)f (are)g Fn(k)f Fr(=)f Fp(C)p Fr(,)j(and)f(Jacobians.)e(W)l(e)j(ha)o(v)o (e)727 1363 y(0)d Fm(!)g Fr(Pic)900 1342 y Fl(0)900 1376 y Fk(k)938 1363 y Fm(!)g Fr(Pic)1072 1370 y Fk(K)1124 1363 y Fm(!)g Fp(Z)0 1472 y Fr(where)k(Pic)216 1451 y Fl(0)216 1485 y Fk(k)259 1472 y Fr(is)g(a)g(compact)g(ab)q(elian)g (group)f(when)h(giv)o(en)f(a)i(decen)o(t)f(top)q(ology)l(.)g(W)l(e)h (can)f(also)g(mak)o(e)0 1542 y(it)f(in)o(to)f(an)g(Ab)q(elian)g(v)m (ariet)o(y)l(.)100 1612 y(A)h(lot)g(of)g(this)g(theory)g(\(except)h (K\177)-25 b(ahler)16 b(di\013eren)o(tials\))g(w)o(as)g(dev)o(elop)q (ed)h(in)f(the)h(30's)g(b)o(y)f(Hasse,)0 1682 y(Artin,)c(and)g (Deuring.)g(They)g(w)o(ere)h(mostly)f(using)f(\014eld)h(theoretic)h (ideas)f({)h(W)l(eil)f(w)o(as)g(the)h(\014rst)f(p)q(erson)0 1751 y(to)k(break)f(a)o(w)o(a)o(y)f(and)h(use)g(geometric)g(ideas.)f (He)i(w)o(as)f(also)g(the)h(\014rst)e(to)i(pro)o(v)o(e)e(Riemann)g(h)o (yp)q(othesis)0 1821 y(for)21 b(curv)o(es.)f(More)h(information)f(on)g (this)h(can)g(b)q(e)h(found)f(in)f(Cornell-Silv)o(erman)e(or)j(an)o(y)g (text)h(on)0 1891 y(Arithmetic)16 b(Geometry)l(.)0 2011 y Fp(Lecture)k(35)c Fr(\(No)o(v)o(em)o(b)q(er)g(17,)g(1995\).)0 2081 y(Scrib)q(e:)g(Ezra)g(Miller)100 2202 y(F)l(or)g(Exercise)i(22,)f (it)h(migh)o(t)f(b)q(e)h(useful)f(to)h(ha)o(v)o(e)g(the)g(follo)o(wing) e(fact.)j(\(Let)g Fn(K)q(=k)g Fr(b)q(e)f(as)f(usual:)0 2271 y(\014nitely)f(generated)g(of)h(transcendence)f(degree)g(=)g(1.\)) 0 2381 y Fp(Theorem)p Fr(:)f(Let)i Fn(P)k Fm(2)14 b Fn(C)480 2388 y Fk(K)518 2381 y Fr(.)i(If)h(deg)f Fn(P)21 b Fr(=)14 b(1)i(then)h Fn(P)23 b Fr(is)16 b(non-singular.)0 2491 y Fp(Corollary)p Fr(:)e(If)g(there)g(is)f(a)h(p)q(oin)o(t)f(of)h (degree)f(1,)g(then)h Fn(K)q(=k)h Fr(is)e(separably)g(generated)g (\(i.e.)h(the)g(generic)0 2560 y(p)q(oin)o(t)i(is)g(non-singular\).)0 2670 y Fp(Question)p Fr(:)g(If)g(deg:Div)468 2677 y Fk(K)520 2670 y Fm(!)e Fp(Z)i Fr(is)f(surjectiv)o(e,)h(do)q(es)g(it)g(follo)o(w) f(that)h Fn(K)q(=k)i Fr(is)d(separably)g(generated?)950 2790 y(71)p eop %%Page: 72 72 72 71 bop 0 50 a Fm(x)19 b Fp(Zeta)h(F)-5 b(unctions)19 b(o)n(v)n(er)h(Finite)h(Fields)0 159 y(Notation)p Fr(:)c Fn(k)32 b Fr(=)16 b(\014nite)g(\014eld)g(of)h(order)e Fn(q)k Fr(\(i.e.)e Fn(k)e Fr(=)e Fp(F)1058 166 y Fk(q)1081 159 y Fr(\))248 229 y Fn(k)274 236 y Fk(n)315 229 y Fr(=)g Fp(F)403 236 y Fk(q)422 227 y Fc(n)464 229 y Fr(=)j(degree)g Fn(n)h Fr(extension)f(of)h Fn(k)248 297 y(K)h Fr(=)e(f.g.)g(extension)h (of)g Fn(k)h Fr(of)e(transcendence)g(degree)g(=)g(1)h(with)1514 284 y(\026)1512 297 y Fn(k)c Fm(\\)e Fn(K)18 b Fr(=)13 b Fn(k)248 367 y(\020)t(;)24 b(\020)334 376 y Fk(K=k)413 367 y Fn(;)g(Z)q(;)h(Z)559 376 y Fk(K=k)652 367 y Fr(=)16 b(zeta)h(functions)f(of)h Fn(K)q(=k)248 437 y(C)284 446
y Fk(K=k)376 437 y Fr(=)d Fn(C)j Fr(=)f(the)h(usual)e(curv)o(e)0 546 y Fp(Lemma)p Fr(:)f(F)l(or)i(eac)o(h)g Fn(n)e Fm(\025)f Fr(0)k(there)f(exist)h(only)f(\014nitely)h(man)o(y)e(divisors)g(of)i (degree)f Fn(n)p Fr(.)100 616 y Fp(Pro)r(of)p Fr(:)d(It)j(su\016ces)e (to)i(consider)e(prime)g(divisors,)g(and)g(the)i(case)f(for)g Fn(K)20 b Fr(follo)o(ws)14 b(from)h(the)g(case)0 686 y(for)i Fn(k)r Fr(\()p Fn(t)p Fr(\),)g(where)g Fn(t)g Fr(is)g(transcenden)o(tal)e(o)o(v)o(er)i Fn(k)r Fr(.)f(No)o(w)h(eac)o (h)g(prime)f(of)h Fn(k)r Fr(\()p Fn(t)p Fr(\))h(of)f(degree)g Fn(n)g Fr(is)g(obtained)0 756 y(from)d(an)g(irreducible)e(p)q (olynomial)i(of)g(degree)g Fn(n)h Fr(o)o(v)o(er)e Fn(k)j Fr(\(except)g(for)e(the)h(single)e(prime)g Fm(1)p Fr(\),)i(and)f(the)0 825 y(result)i(is)g(no)o(w)g(ob)o(vious.)0 935 y Fp(De\014nition)p Fr(:)i Fn(Z)313 944 y Fk(K=k)405 935 y Fr(=)c Fn(Z)492 944 y Fk(K=k)571 935 y Fr(\()p Fn(T)7 b Fr(\))14 b Fm(2)g Fn(Z)t Fr([[)p Fn(T)7 b Fr(]])13 b Fm(\032)h Fp(C)p Fr(\(\()p Fn(T)7 b Fr(\)\))18 b(is)e(de\014ned)g(as)723 1038 y Fh(X)663 1145 y Fk(D)q Fj(2)p Fl(Div)787 1152 y Fc(K=k)717 1188 y Fk(D)q Fj(\025)p Fl(0)869 1086 y Fn(T)912 1065 y Fl(deg)7 b Fk(D)1029 1086 y Fr(=)1099 1023 y Fj(1)1083 1038 y Fh(X)1082 1144 y Fk(n)p Fl(=0)1165 1086 y Fn(A)1202 1093 y Fk(n)1229 1086 y Fn(T)1265 1065 y Fk(n)0 1285 y Fr(where)659 1354 y Fn(A)696 1361 y Fk(n)738 1354 y Fr(=)13 b(#)840 1314 y Fh(\010)869 1354 y Fn(D)j Fm(\025)d Fr(0)1017 1312 y Fh(\014)1017 1342 y(\014)1047 1354 y Fr(deg)q Fn(D)i Fr(=)f Fn(n)1262 1314 y Fh(\011)0 1449 y Fr(and)i Fn(A)134 1456 y Fl(0)170 1449 y Fr(=)e(1.)0 1559 y Fp(Example)p Fr(:)i Fn(K)i Fr(=)13 b Fn(k)r Fr(\()p Fn(t)p Fr(\).)k(If)g(w)o(e)f(ha)o(v)o(e)487 1674 y Fn(D)24 b Fr(=)613 1627 y Fh(X)633 1734 y Fk(P)693 1674 y Fn(n)723 1681 y Fk(P)767 1674 y Fm(\001)11 b Fn(P)29 b Fr(=)22 b Fn(n)944 1681 y Fj(1)997 1674 y Fm(\001)11 b(1)28...