Number Theory: A Seminar held at the Graduate School and by Richard T. Bumby (auth.), David V. Chudnovsky, Gregory V.

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

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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

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1. W e def let P0 (x0) = Q (x0), and define rational functions Lp, j(p,j=0, .... Pp (x0) = 6i,j , i,j = 0 ..... 1. 35 If we put v(x0) d~f det(Pj (x0): i,j = 0,I ..... 1 and Lp, j = Rp, j(x0)/V(x0), from P~-i>(x0) , q q / p constructed Let ~ = min{ordx0=0(P 0 where Rp, j(x0) by Cramer's is a determinant rule from equations (x0)) , i = 0 ..... n]. 1). ,n and i = 0, i .... ,n. 2) ordx0=0(Rp,j(x0)) n ~ ~ + (n-l) (M+D+I) - ~i=2 ki' (p,j = 0 ..... n) . Let us denote, wj = for simplicity, f0(x0) -- l0 If we put zn p=0 Lp, j-fp(X0): j = 0 .....

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(x) of the coefficients bounded by C ~ kd. Let (Q(x);Pl(X), ....

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].

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