p-adic Differential Operators on Automorphic Forms on Unitary Groups
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 177-243.

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 -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)×U(n).

Nous construisons certains opérateurs différentiels C 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 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)×U(n).

DOI: 10.5802/aif.2704
Classification: 14G35, 11G10, 11F03, 11F55, 11F60
Keywords: $p$-adic automorphic forms, differential operators, Maass operators
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, Volume 62 (2012) no. 1, pp. 177-243. doi : 10.5802/aif.2704. http://www.numdam.org/articles/10.5802/aif.2704/

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