Special values of Goss $L$-series attached to Drinfeld modules of rank 2

Gezmiş, Oğuz

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 2, pp. 511-552.

Inspiré par le cadre classique, Goss a défini des séries $L$ attachées aux modules de Drinfeld. Dans cet article, pour une puissance fixée $q$ d’un nombre premier et un module de Drinfeld donné $\varphi$ de rang 2 avec une certaine condition sur ses coefficients, nous donnons des formules explicites pour les valeurs de la série $L$ de Goss attachée à $\varphi$ aux entiers positifs $n$ tels que $2n+1\le q$ en termes de polylogarithmes et coefficients de la série logarithmique de $\varphi$.

Inspired by the classical setting, Goss defined $L$-series attached to Drinfeld modules. In this paper, for a fixed choice of a power $q$ of a prime number and a given Drinfeld module $\varphi$ of rank 2 with a certain condition on its coefficients, we give explicit formulas for the values of Goss $L$-series attached to $\varphi$ at positive integers $n$ such that $2n+1\le q$ in terms of polylogarithms and coefficients of the logarithm series of $\varphi$.

Classification : 11G09, 11M38
Mots clés : Drinfeld modules, $L$-series, $t$-modules

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