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Dr. Katz DEq Homework Solutions 85

# Dr. Katz DEq Homework Solutions 85 - finite ~tale covering...

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Algebraic Solutions of Differential Equations 85 6.1. The Hypergeometric Equation-Definition (6.1.0) For any scheme T, we will denote by 2 the standard coordinate on A~, and by S r the open subset of A~ where the section 2(1-2)e F(A~, (9A~) is invertible. For any sections a,b,c~F(T, Or), we define the hypergeometric module E(a, b, c) on Sr to be the free (9sT-module of rank two with base eo, el, and integrable T-connection (6.1.1) ( ff_~) (c-(a+b+ l)2) ab V (e0 = 2(1-2) el q A(1-2) e~ As explained in (6.0.4.3), the horizontal sections of the dual of E(a, b, c) over an open set q/of ST "are" the sections f~F(~l, (9~u) which satisfy the hypergeometric equation with parameters a, b, c (6.1.2) d 2 ~-abf =O. 6.2. Theorem. Let T= Spec(A), with A a finitely generated subring of C, and let a, b, c~A. The following conditions are equivalent: (6.2.1) The hypergeometric module E(a, b, c) on S c becomes trivial on a
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Unformatted text preview: finite ~tale covering of S c. (6.2.2) There exists a finite extension of the field C(2) over which E(a, b,c) becomes trivial. (6.2.3) The hypergeometric equation (6.1.2) with parameters a, b, c has a full set of algebraic solutions (cf. (5.4.2.1)). (6.2.4) The hypergeometric equation (6.1.2) with parameters a, b, c, has a finite monodromy group. (6.2.5) The inverse image of E (a, b, c) on C0.)/C has p-curvature zero for almost all primes p (cf. (5.4.3.2)). (6.2.6) The parameters a, b, c are all rational numbers, and for almost all primes p, the reduction modulo p of the hypergeometric equation (6.1.2) with parameters a, b, c has two solutions in Fp [2] (resp. Fp (2), resp. Fp[[2]], resp. Fp((2))) which are linearly independent over Fp(2 p) (resp. F(2P), resp. Fp(2"), resp. Fp((2P)))....
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