The goal of this paper is to study certain -adic differential operators on automorphic forms on . These operators are a generalization to the higher-dimensional, vector-valued situation of the -adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the -adic case of the -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 -adic -functions attached to -adic families of automorphic forms on the unitary groups .
Nous construisons certains opérateurs différentiels et leurs analogues -adiques, qui agissent sur des formes automorphes (à valeurs vectorielles ou scalaires) pour les groupes unitaires . Nous étudions des propriétés de ces opérateurs, et nous les utilisons à prouver quelques théorèmes arithmetiques. Ces opérateurs différentiels sont une généralisation au cas -adique des opérateurs différentiels 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 -adiques à valeurs vectorielles construits pour les formes modulaires par N. Katz. Ils devraient être utiles dans la construction de certaines fonctions -adiques, en particulier les fonctions -adiques attachées aux familles -adiques de formes automorphes pour les groupes unitaires .
Mots-clés : $p$-adic automorphic forms, differential operators, Maass operators
@article{AIF_2012__62_1_177_0, author = {Eischen, Ellen E.}, title = {$p$-adic {Differential} {Operators} on {Automorphic} {Forms} on {Unitary} {Groups}}, journal = {Annales de l'Institut Fourier}, pages = {177--243}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {62}, number = {1}, year = {2012}, doi = {10.5802/aif.2704}, zbl = {1257.11054}, mrnumber = {2986270}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2704/} }
TY - JOUR AU - Eischen, Ellen E. TI - $p$-adic Differential Operators on Automorphic Forms on Unitary Groups JO - Annales de l'Institut Fourier PY - 2012 SP - 177 EP - 243 VL - 62 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2704/ DO - 10.5802/aif.2704 LA - en ID - AIF_2012__62_1_177_0 ER -
%0 Journal Article %A Eischen, Ellen E. %T $p$-adic Differential Operators on Automorphic Forms on Unitary Groups %J Annales de l'Institut Fourier %D 2012 %P 177-243 %V 62 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2704/ %R 10.5802/aif.2704 %G en %F AIF_2012__62_1_177_0
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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