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].
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Submitted on : Thursday, March 28, 2013 - 1:38:42 PM
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  • HAL Id : hal-00776680, version 2
  • ARXIV : 1301.3608

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

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