We give a proof of the existence of Asai, exterior square, and symmetric square local -functions, -factors and root numbers in characteristic – the case of 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 -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 . Furthermore, in order to be self contained, we include a treatise of -functions arising from maximal Levi subgroups of general linear groups.
Nous donnons une preuve de l’existence des fonctions locales d’Asai, extérieur et symétrique carré ainsi que des facteurs et correspondants en caractéristique – le cas 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 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 . De plus, afin d’être complet, nous décrivons les fonctions provenant d’un sous-groupe maximal de Levi d’un groupe linéaire général.
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Keywords: Automorphic $L$-funcitons, functional equation, Langlands-Shahidi method, local factors
Mot clés : Fonctions $L$-automorphes, équation fonctionnelle, méthode Langlands-Shahidi, facteurs locaux
Lomelí, Luis Alberto 1
@article{AIF_2016__66_5_1733_0, author = {Lomel{\'\i}, Luis Alberto}, title = {On automorphic $L$-functions in positive characteristic}, journal = {Annales de l'Institut Fourier}, pages = {1733--1771}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {66}, number = {5}, year = {2016}, doi = {10.5802/aif.3049}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3049/} }
TY - JOUR AU - Lomelí, Luis Alberto TI - On automorphic $L$-functions in positive characteristic JO - Annales de l'Institut Fourier PY - 2016 SP - 1733 EP - 1771 VL - 66 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3049/ DO - 10.5802/aif.3049 LA - en ID - AIF_2016__66_5_1733_0 ER -
%0 Journal Article %A Lomelí, Luis Alberto %T On automorphic $L$-functions in positive characteristic %J Annales de l'Institut Fourier %D 2016 %P 1733-1771 %V 66 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3049/ %R 10.5802/aif.3049 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.3049/
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