92
N.M. Katz:
By (6.4.0), this implies that the hypergeometric equation with para
meters a, b, c has "two" mod p solutions for all sufficiently large primes
p satisfying
pA
 1 (N). As there are only a finite number of d's to consider
(Z/NZ being finite), (6.2.6) follows. Now for the necessity. Choose a A
invertible in
Z/NZ.
By hypothesis, for every sufficiently large prime p
with
pA
 1 (N), we have either
(6.6.0.4)
1 Rp(b)>lRp(C)>= l Rp(a)
P
P
P
or
(6.6.0.5)
1 Rp( a) > 1 Rp( c) >I Rp( b).
P
P
P
As there are infinitely many primes with
p AI(N),
either (6.6.0.4) or
(6.6.0.5) must hold for infinitely many such primes. As the roles of a and
b are symmetric, we may suppose (6.6.0.4) holds infinitely often for such
primes. We may then apply the limit formula (6.5.3.1) to (6.6.0.4), to
obtain one of the following inequalities
(6.6.0.6)
l>(bA)>(cA)>(aA)>O
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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