By Richard T. Bumby (auth.), David V. Chudnovsky, Gregory V. Chudnovsky, Harvey Cohn, Melvyn B. Nathanson (eds.)

**Read Online or Download Number Theory: A Seminar held at the Graduate School and University Center of the City University of New York 1983–84 PDF**

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**Extra info for Number Theory: A Seminar held at the Graduate School and University Center of the City University of New York 1983–84**

**Example text**

1. W e

4 and Theorem According integers value at most C ~ ~. a nonzero in of all coef- M = [(i/n - 8)D]. 1, AD+M. [(i/n-8)D] D + 1 unknowns Lemma [Q. fi]D, of the second kind of fl(x), .... fn(X) with parameters Pi(x)) ~ D + [(i/n - 8)D] + 1 for all i = 1 .... ,n. 2) of differential equations with lam, il ~ c 1 (m = 0,i ..... ,n) for some C 1 > I. be the system of Pad~ approximants (D,D,[(I/n - ~)D]) constructed kd < [(i/n - 6)D], then Q

L)p is Fuchsian. I) are Fuchsian. i) is Fuchsian. i) has sufficiently many solutions is a weak sense for almost all ~, has all exponents at regular singularities, being rational numbers. To prove this result we need one form of Chebotarev's density theorem that will be proved below by means of Pad~ approximations. This version (or rather a corollary of) Chebotarev's density theorem is itself equivalent to the simplest case of the G r o t h e n d i e c k conjecture, as was d e m o n s t r a t e d by Honda [2].