Dr. Katz DEq Homework Solutions 92

Dr. Katz DEq Homework Solutions 92 - 92 N.M. Katz: B y...

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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 A--I(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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