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Dr. Katz DEq Homework Solutions 86

Dr. Katz DEq Homework Solutions 86 - 86 N.M Katz P roof...

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86 N.M. Katz: Proof By (5.4.1), (6.2.1).r and by (6.0.4.3), (6.2.2)r Be- cause the hypergeometric equation has regular singular points, (6.2.3)~ (6.2.4). By (5.4.4), we have (6.2.2)~ (6.2.5). By ([24], Theorem 1.3.0), (6.2.5) implies that E(a, b, c) has regular singular points on Sc, and that its local monodromy around each singular point 0, 1, oo is of finite order. In particular, (6.2.5) implies that E(a, b, c) on Sc has rational exponents at each of 0, 1, ~, or, what is the same, that the hypergeometric equa- tion (6.1.2) with parameters a, b, c has rational exponents at 0, 1, ~. As the exponents are 0 and 1 -c at 0, 0 and c- a- b at 1, and a and b at ~, (6.2.5) implies that a, b, c~Q. The rest of the implication (6.2.5)~ (6.2.6) results from the fact that E(a, b, c) "comes from" T-an open subset of Spec(Z), so that we may test for "p-curvature zero for almost all p" by seeing if the reduction modp on E(a, b, c) on Svp has p-curvature zero for almost all p (and perform this latter test prime by prime availing ourselves of (6.0.6.7) and (6.0.7)).
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