$p$-adic Differential Operators on Automorphic Forms on Unitary Groups

Eischen, Ellen E.

doi:10.5802/aif.2704

p

-adic Differential Operators on Automorphic Forms on Unitary Groups
[Opérateurs différentiels

p

-adiques sur formes automorphes pour groupes unitaires]

Eischen, Ellen E.¹

¹ Mathematics Department Northwestern University 2033 Sheridan road Evanston, IL 60208 USA

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 177-243.

Résumé
Abstract

Nous construisons certains opérateurs différentiels $C^{\infty}$ et leurs analogues $p$ -adiques, qui agissent sur des formes automorphes (à valeurs vectorielles ou scalaires) pour les groupes unitaires $U (n, n)$ . Nous étudions des propriétés de ces opérateurs, et nous les utilisons à prouver quelques théorèmes arithmetiques. Ces opérateurs différentiels sont une généralisation au cas $p$ -adique des opérateurs différentiels $C^{\infty}$ étudiés d’abord par H. Maass et étudiés ensuite en détail par M. Harris et G. Shimura. Ils sont une généralisation au cas des opérateurs différentiels $p$ -adiques à valeurs vectorielles construits pour les formes modulaires par N. Katz. Ils devraient être utiles dans la construction de certaines fonctions $L$ $p$ -adiques, en particulier les fonctions $L$ $p$ -adiques attachées aux familles $p$ -adiques de formes automorphes pour les groupes unitaires $U (n) \times U (n)$ .

The goal of this paper is to study certain $p$ -adic differential operators on automorphic forms on $U (n, n)$ . These operators are a generalization to the higher-dimensional, vector-valued situation of the $p$ -adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the $p$ -adic case of the $C^{\infty}$ -differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain $p$ -adic $L$ -functions attached to $p$ -adic families of automorphic forms on the unitary groups $U (n) \times U (n)$ .

EuDML MR Zbl

DOI : 10.5802/aif.2704

Classification : 14G35, 11G10, 11F03, 11F55, 11F60
Mots clés : $p$-adic automorphic forms, differential operators, Maass operators

Affiliations des auteurs :

Eischen, Ellen E. ¹

¹ Mathematics Department Northwestern University 2033 Sheridan road Evanston, IL 60208 USA

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Eischen, Ellen E. $p$-adic Differential Operators on Automorphic Forms on Unitary Groups. Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 177-243. doi : 10.5802/aif.2704. https://aif.centre-mersenne.org/articles/10.5802/aif.2704/

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