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

[Opérateurs différentiels $p$-adiques sur formes automorphes pour groupes unitaires]

Eischen, Ellen E.

Annales de l'Institut Fourier, Tome 62 (2012), p. 177-243 / Harvested from Numdam

Résumé

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 $U (n)\times U (n)$.

Abstract

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 $U(n)\times U(n)$.

Détail

Publié le : 2012-01-01

MR 2986270 | Zbl 1257.11054 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2704
Classification:  14G35,  11G10,  11F03,  11F55,  11F60

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