Dr. Katz DEq Homework Solutions 36

# Dr. Katz DEq Homework Solutions 36 - 36 N.M Katz Then the...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 36 N.M. Katz: Then the Hass-W~tt matrix is given by (2.3.7.16) H-W(F*s(m(- W)))= Z A pw_v m ( - V). (2.3.7.17) Special Case. Hypotheses as in (2.3.7.14), suppose that X has d egree d = n+2. Then R"f, ((gx) is free of rank one on S, with base m ( - 1 , - 1 .... , - 1 ) and the Hasse-Witt matrix of X /S i n dimension n ( or the Hasse invariant, as we shall call it in this case) with respect to the b asis m ( - 1, - 1..... - 1) is given by the coefficient of (X2... X,+2) p-2 i n H p- 2. W e now apply this to a particularly beautiful family of hypersurfaces. (2.3.7.18) Corollary. Let d>2 be relatively prime to p, and put S= S pec(Fp [2] [1/1 - 2d]). Consider the smooth (over S) hypersurface X in Ps o f equation d-1 d ~, X d - d 2 X1... Xd. (2.3.7.19) i =1 It's Hasse invariant is given by the truncated hypergeometrie series (2.3.7.20) (d2)P- 2 ~, (1/d)a(2/d)a...(d-1/d), 2_ad a ! a !... a ! a~O where for ~4=0, we put ( ~)o=1, ( ~ ) , = ~ ( ~ + l ) . . . ( ~ + n - 1 ) /f n > l . This may be expressed in terms of the "full" hypergeometrieseries in Z p [[2- 2]] [2] by means of Dwork's congruence (cf. [8 a], pp. 36-37) (2.3.7.21) (d2) p-2 y' (1/d)a(2/d),...(d- 1/d)a 2_ad=_G(2)/G(2P)modulo(p) a !a!...a! a=o where G (2)=(d2) - 1 F [ 1/d, 2/d..... d - 1/d. given by \ 1, ..., 1 , 2 -d) is the element o f Zp[[2-1]] (2.3.7.22) G(2)=2-1a~o(-l) a'd-1) ( - 1 / d ) ( - 2 / d ) . . . (-(d-a l )/d)2_ad" Proof B y direct calculation, one finds that (2.3.7.20) is t he coefficient o f (l--I x,) v-1 in ~ X ( - d 2 I-I X,. T o apply the congruences of Dwork, i i w e need only observe that in the sum (2.3.7.20), we could have let a run f rom 0 all the way to p - 1 (which is the usual f irst truncation point for h ypergeometric-type series), because, for a ~ p - l b ut a d>p, w e have (2.3.7.23) (ad)! (1/d)a(2/d)a...(d-1/d)a= dad.a! - 0 m odulo p. Q.E.D. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online