Dr. Katz DEq Homework Solutions 103

Dr. Katz DEq Homework Solutions 103 - prime p...

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Algebraic Solutions of Differential Equations 103 is an integer, and that for every integer C > 1 which is invertible modulo n, the numbers Ca~n, (b/n, Cc/n also satisfy (6.9.0) (cf. (6.6.2)). By (6.6.3) and (6.7.1), the hypothesis and conclusion of the alleged implication (6.2.6)~ (6.2.4) depend only on the classes modulo Z of a/n, b/n, c/n. Thus we may and will suppose in addition that a c a (6.9.2.2) -->0, ----->0, 1 +b/n-c/n>O. n n n We define strictly positive integers A, B, C by A c-a B b-c C a (6.9.2.3) - , -1-~ , - . n n n n n n c-b b The non-integrality of c-a, , a , (cf. (6.9.1.1), (6.9.2.1)) implies n n n n that n does not divide any of the integers A, B, C, A + B + C. Then we may apply (6.8.6): (6.9.3) For any integer C > 1 which is invertible modulo n, (6.9.3.0) E((a/n, (b/n, (c/n)lC(2) ~, P (Z (- C)) H~R(X(n; A, B, C)/C(2)). Thus, for every integer 1 < (< n- 1 which is invertible modulo n, (6.9.3.1) P (Z(- C)) Hi, (X (n; A, B, C)/C (2)) with its Gauss-Manin connection has p-curvature zero for almost all
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Unformatted text preview: prime p (cf.(5.4.3.3)). As C varies over the integers l&lt;C&lt;n-1, the characters Z(-C) run over a Q-conjugacy class of irreducible repre- sentations of#, (namely, the faithful ones). Thus we may apply Theorem 5.7 to deduce that for every integer C &gt;__ 1 invertible modulo n, (6.9.3.2) P(X(-C)) H~,(X (n; A, B, C)) with the Gauss-Manin connection has a full set of algebraic solutions. By (6.9.3.0), it follows that for every integer C&gt; 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 &gt;_ 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 &lt;- C &lt;- n- 1...
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