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Unformatted text preview: 36 N.M. Katz: Then the Hass-W~tt matrix is given by
(184.108.40.206) H-W(F*s(m(- W)))= Z A pw_v m ( - V). (220.127.116.11) Special Case. Hypotheses as in (18.104.22.168), 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.
(22.214.171.124) Corollary. Let d>2 be relatively prime to p, and put S=
S pec(Fp  [1/1 - 2d]). Consider the smooth (over S) hypersurface X in
Ps o f equation
d ~, X d - d 2 X1... Xd. (126.96.36.199) i =1 It's Hasse invariant is given by the truncated hypergeometrie series
(188.8.131.52) (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]] 
by means of Dwork's congruence (cf. [8 a], pp. 36-37) (184.108.40.206) (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.
1, ..., 1
, 2 -d) is the element o f Zp[[2-1]]
(220.127.116.11) 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 (18.104.22.168) 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 (22.214.171.124), 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
(1/d)a(2/d)a...(d-1/d)a= dad.a! - 0 m odulo p. Q.E.D. ...
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- Fall '11