Universal Gauss-Thakur sums and L-series

TL;DR

This paper explores the properties of the omega function in Anderson-Thakur theory, revealing its role as a universal Gauss-Thakur sum, and applies these findings to L-series relations and Bernoulli-Carlitz fractions.

Contribution

It demonstrates that omega and its derivatives generate the maximal abelian extension of F_q(T) and establishes functional relations for certain L-series.

Findings

omega generates the maximal abelian extension of F_q(T)

Functional relations for specific L-series are proven

New congruences for Bernoulli-Carlitz fractions are derived

Abstract

In this paper we study the behavior of the function omega of Anderson-Thakur evaluated at the elements of the algebraic closure of the finite field with q elements F_q. Indeed, this function has quite a remarkable relation to explicit class field theory for the field K=F_q(T). We will see that these values, together with the values of its divided derivatives, generate the maximal abelian extension of K which is tamely ramified at infinity. We will also see that omega is, in a way that we will explain in detail, an universal Gauss-Thakur sum. We will then use these results to show the existence of functional relations for a class of L-series introduced by the second author. Our results will be finally applied to obtain a new class of congruences for Bernoulli-Carlitz fractions, and an analytic conjecture is stated, implying an interesting behavior of such fractions modulo prime ideals of A=F_q[T].