Dr. Katz DEq Homework Solutions 36

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

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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. ...
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