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.


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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