Universal Gauss-Thakur sums and L-series
Bruno Angles (LMNO), Federico Pellarin (ICJ)

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…
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research
