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

Oğuz Gezmiş

doi:10.5802/jtnb.1168

Special values of Goss

L

-series attached to Drinfeld modules of rank 2

Oğuz Gezmiş ¹

¹ National Center for Theoretical Sciences National Taiwan University Taipei, Taiwan R.O.C.

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

Abstract
Résumé

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 $ϕ$ of rank 2 with a certain condition on its coefficients, we give explicit formulas for the values of Goss $L$ -series attached to $ϕ$ at positive integers $n$ such that $2 n + 1 \leq q$ in terms of polylogarithms and coefficients of the logarithm series of $ϕ$ .

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 $ϕ$ 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 $ϕ$ aux entiers positifs $n$ tels que $2 n + 1 \leq q$ en termes de polylogarithmes et coefficients de la série logarithmique de $ϕ$ .

Received: 2020-05-28
Revised: 2020-10-06
Accepted: 2021-06-05
Published online: 2021-09-10

DOI: 10.5802/jtnb.1168

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

Author's affiliations:

Oğuz Gezmiş ¹

¹ National Center for Theoretical Sciences National Taiwan University Taipei, Taiwan R.O.C.

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {O\u{g}uz Gezmi\c{s}},
     title = {Special values of {Goss} $L$-series attached to {Drinfeld} modules of rank 2},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {511--552},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Oğuz Gezmiş. Special values of Goss $L$-series attached to Drinfeld modules of rank 2. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 511-552. doi : 10.5802/jtnb.1168. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1168/

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