Universal Gauss-Thakur sums and L-series
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].
Keywords
Cite
@article{arxiv.1301.3608,
title = {Universal Gauss-Thakur sums and L-series},
author = {Bruno Angles and Federico Pellarin},
journal= {arXiv preprint arXiv:1301.3608},
year = {2013}
}
Comments
Corrected several typos and an error in the proof of Proposition 21 Section 3. Improved the general presentation of the paper