Unformatted text preview: 36 N.M. Katz: Then the HassW~tt matrix is given by
(2.3.7.16) HW(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 HasseWitt 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) p2
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
d1
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...(d1/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. 3637) (2.3.7.21) (d2) p2 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[[21]]
(2.3.7.22) G(2)=21a~o(l) a'd1) (  1 / d ) (  2 / d ) . . . ((da l )/d)2_ad" Proof B y direct calculation, one finds that (2.3.7.20) is t he coefficient
o f (lI x,) v1 in ~ X (  d 2 II 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 ypergeometrictype 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...(d1/d)a= dad.a!  0 m odulo p. Q.E.D. ...
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 Fall '11
 NormanKatz
 Prime number, Mathematical terminology, D2, Congruence relation, Truncation, Dwork

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