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Unformatted text preview: 30 N.M. Katz: is a s ection o f the inclusion F~= i r
~1, whose kernel is none other
t hen F i+lnF~o~ ~1. T his shows that the canonical mapping F~+lc~
F~o~i  1 ~ F~o~ia/Ffo~iis a n isomorphism, which proves (2.3.4.4.2).
(2.3.5) In this section we wish to explain the differential equations
satisfied by certain of the higher HasseWitt matrices introduced in
(2.3.4). These differential equations were first noticed by Igusa [22] in
t he case of elliptic curves, then later explained quite generally by Manin
[28].
(2.3.5.1) We place ourselves under the hypotheses of (2.3.2), and assume
f urther that the divisor D is void, and that the geometric fibres of X /S a re
c onnected, of dimension N. Under these hypotheses, we have
(2.3.5.1.1) R2Nf,(~Q~c/s)%RNf,(O~/s) is locally free of rank 1 and even
c anonically isomorphic to the structural sheaf Cs, via the trace morphism
(cf. [15] and [42])
tr: RaNf , (Y2]/S)~ d)s (2.3.5.1.2) w hich carries the GaussManin connection on R ZNf,(f2]/s) i nto the
s tandard connection on d)s (the one given by exterior differentiation
d : (9s'~ 12~/r).
T he cupproduct pairings
(2.3.5.1.3) Rn f , ([2*x/s)@ R 2N'f, (fd~c/s)~ tr R2N f (Q~c/s) " ' (gs a re perfect dualities of coherent locally free S modules, for which the
f iltrations F and F~o~are both selfdual in the sense that (2.3.5.1.4) ( F' R n f , (O~/s)) • = F N +1 n R a N  nf , (f2~/s) (2.3.5.1.5) (F~'onR ' f , (O~c/s))• = F~o, ~n RaNn f , (12~;/s).
+ T he associated graded pairings
Ni
2Nn
r iR n 12*
f2*
g F( f*(X/S))  gF (R
f,(x/s) ni i N+in ~ N 2N 9 r
gF( R f,( [2 s))) Ni a nd
, grfoo. (R2Sf, (f2~:/s))
(2.3.5.1.7)
F ~* i
,bs(Rn   i f ,(~x/s)) /l /l
* N+in Nf,(f2x/si )) ~ *
N
. F.bsR N f,(f2x/s) "" __~(9
s ...
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 Fall '11
 NormanKatz

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