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Transcendence and special zeta values in characteristic p. (English) Zbl 0734.11040

Let \(A={\mathbb{F}}_ q[T]\) and \(K={\mathbb{F}}_ q(T)\) be the ring of polynomials, field of rational functions over a finite field \({\mathbb{F}}_ q\), respectively, and let \(\bar K_{\infty}\) be the algebraic closure of the \(\infty\)-adic completion \(K_{\infty}\) of K. L. Carlitz and D. Goss introduced the \(\bar K_{\infty}\)-valued zeta function \(\zeta\) (s) of A, whose values at positive integers \(s=k\) are given by \(\sum \epsilon n^{-k}\) (n\(\in A\) monic). This zeta function, as well as its special values and v-adic analogues \(\zeta_ v(s)\) (v running through the places of A) are related to the class field theory of K. If the integer k is divisible by q-1, \(\zeta\) (k) is a product of a Bernoulli-like “number” \(B_ k\in K\) and \({\tilde \pi}{}^ k\), where \({\tilde \pi}\) is a fundamental period of the Carlitz module, i.e., plays the role of classical \(2\pi\) i. Wade proved its transcendence over K in 1941. Now the amazing thing is that transcendence properties of \(\zeta\) (k) and \(\zeta_ v(k)\) may also be proved for k not divisible by q-1, which corresponds to classical odd zeta values.
This is done in the paper, using earlier results of the author and recent work of Anderson and Thakur.

MSC:

11J89 Transcendence theory of elliptic and abelian functions
11T55 Arithmetic theory of polynomial rings over finite fields
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
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