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Unformatted text preview: prime p (cf.(5.4.3.3)). As C varies over the integers l<C<n1, the characters Z(C) run over a Qconjugacy class of irreducible repre sentations of#, (namely, the faithful ones). Thus we may apply Theorem 5.7 to deduce that for every integer C >__ 1 invertible modulo n, (6.9.3.2) P(X(C)) H~,(X (n; A, B, C)) with the GaussManin connection has a full set of algebraic solutions. By (6.9.3.0), it follows that for every integer C> 1 which is invertible modulo n, in particular for C = 1, the hypergeometric module (6.9.3.3) E (C a/n, C b/n, C c/n)[ C (2) has a full set of algebraic solutions. Q.E.D. (6.9.4) Corollary. Let a, b, c, n be integers, n >_ 1, and suppose none of the exponent d!fference (6.9.4.0) 1  c/n, c/n  a/n  b/n, a/n  b/n is an integer. Then the hypergeometric equation with parameters a/n, b/n, c/n has two algebraic solutions ![ and only if, for every integer 1 < C < n 1...
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 Fall '11
 NormanKatz

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