On automorphic L-functions in positive characteristic
Annales de l'Institut Fourier, Volume 66 (2016) no. 5, p. 1733-1771

We give a proof of the existence of Asai, exterior square, and symmetric square local L-functions, γ-factors and root numbers in characteristic p – the case of p=2 included. Our study is made possible by developing the Langlands-Shahidi method over a global function field in the case of a Siegel Levi subgroup of a split classical group or a quasi-split unitary group. The resulting automorphic L-functions are shown to satisfy a rationality property and a functional equation. A uniqueness result of G. Henniart and the author allows us to show that the definitions provided in this article are in accordance with the local Langlands conjecture for GL n . Furthermore, in order to be self contained, we include a treatise of L-functions arising from maximal Levi subgroups of general linear groups.

Nous donnons une preuve de l’existence des fonctions L locales d’Asai, extérieur et symétrique carré ainsi que des facteurs γ et ε correspondants en caractéristique p – le cas p=2 étant inclus. Notre étude est possible grâce à la méthode de Langlands-Shahidi sur un corps de fonctions global dans le cas d’un sous-groupe de Siegel Levi d’un groupe classique déployé ou d’un groupe unitaire quasi-déployé. Les fonctions L qui en résultent satisfont une propriété de rationalité et une équation fonctionnelle. Un résultat d’unicité de G. Henniart et de l’auteur permet de montrer que les définitions données dans cet article sont compatibles avec la conjecture de Langlands locale pour GL n . De plus, afin d’être complet, nous décrivons les fonctions L provenant d’un sous-groupe maximal de Levi d’un groupe linéaire général.

Received : 2012-11-27
Revised : 2015-11-18
Accepted : 2016-03-03
Published online : 2016-07-28
DOI : https://doi.org/10.5802/aif.3049
Classification:  11F70,  11M38,  22E50,  22E55
Keywords: Automorphic L-funcitons, functional equation, Langlands-Shahidi method, local factors
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     author = {Lomel\'\i , Luis Alberto},
     title = {On automorphic $L$-functions in  positive characteristic},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     pages = {1733-1771},
     doi = {10.5802/aif.3049},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_5_1733_0}
}
Lomelí, Luis Alberto. On automorphic $L$-functions in  positive characteristic. Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 1733-1771. doi : 10.5802/aif.3049. http://www.numdam.org/item/AIF_2016__66_5_1733_0/

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