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Differential operators, pullbacks, and families of automorphic forms on unitary groups

Content Provider	Springer Nature Link
Author	Eischen, Ellen Elizabeth
Copyright Year	2016
Abstract	This paper has two main parts. First, we construct certain differential operators, which generalize operators studied by G. Shimura. Then, as an application of some of these differential operators, we construct certain p-adic families of automorphic forms. Building on the author’s earlier work, these differential operators map automorphic forms on a unitary group of signature (n, n) to (vector-valued) automorphic forms on the product $$U^\varphi \times U^{-\varphi }$$ of two unitary groups, where $$U^\varphi $$ denotes the unitary group associated to a Hermitian form $$\varphi $$ of arbitrary signature on an n-dimensional vector space. These differential operators have both a p-adic and a $$C^{\infty }$$ incarnation. In the scalar-weight, $$C^{\infty }$$ -case, these operators agree with ones studied by Shimura. In the final section of the paper, we also discuss some generalizations to other groups and settings. The results from this paper apply to the author’s paper-in-preparation with J. Fintzen, E. Mantovan, and I. Varma and to her ongoing joint project with M. Harris, J. -S. Li, and C. Skinner; they also relate to her recent paper with X. Wan.Cet article se divise principalement en deux parties. Tout d’abord. nous contrui-sons des opérateurs différentiels généralisant les opérateurs étudiés par G. Shimura. Puis, nous appliquons certains de ces opérateurs différentiels pour construire quelques familles p-adiques de formes automorphes. Il s’ensuit des travaux antérieurs de l’auteur que ces opérateurs différentiels envoient les formes automorphes sur le groupe unitaire de signature (n, n) sur des formes automorphes (à valeurs vectoriels) sur le produit de deux groupes unitaires, $$U^{\varphi }\times U^{-\varphi }$$ , où $$U^{\varphi }$$ désigne le groupe unitaire associé à une forme hermitienne de signature arbitraire sur un espace vectoriel de dimension n. Ces opérateurs différentiels ont à la fois une incarnation p-adique et une incarnation $$C^{\infty }$$ . Dans le cas $$C^{\infty }$$ et lorsque le poids est scalaire, ces opérateurs correspondent à ceux étudiés par Shimura. Dans la dernière partie de l’article, nous abordons quelques généralisations à d’autres groupes et situations. Les résultats de cet article s’appliquent à l’article de J. Fintzen, E. Mantovan, I. Verma et l’auteur, actuellement en préparation, et au project qu’elle mène conjointement avec M. Harris, J.-S. Li et C. Skinner; ils sont également en lien avec son article récent avec X. Wan.
Starting Page	55
Ending Page	82
Page Count	28
File Format	PDF
ISSN	21954755
Journal	Annales mathématiques du Québec
Volume Number	40
Issue Number	1
e-ISSN	21954763
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-03-03
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Other groups and their modular and automorphic forms p-adic automorphic forms Modular and automorphic functions Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms Mathematics Congruences for modular and $p$-adic modular forms $p$-adic theory, local fields Differential operators Algebra Analysis Forms of half-integer weight; nonholomorphic modular forms Number Theory Shimura varieties
Content Type	Text
Resource Type	Article
Subject	Mathematics

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