Universal Gauss-Thakur sums and L-series

Bruno Angles (1) , Federico Pellarin (2)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 CTN - Combinatoire, théorie des nombres

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].

Mots clés: Function fields of positive characteristic; Anderson-Thakur function; Abelian extensions; $L$-functions in positive characteristic

Domaines: Théorie des nombres [math.NT]

Bruno Angles, Federico Pellarin. Universal Gauss-Thakur sums and L-series. Inventiones Mathematicae, 2015, 200 (2), pp.653-669. ⟨hal-00776680v2⟩