Annales mathématiques du Québec

, Volume 40, Issue 1, pp 55–82 | Cite as

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

  • Ellen Elizabeth Eischen


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 (nn) 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.


p-adic automorphic forms Differential operators Shimura varieties 


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 (nn) 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.

Mathematics Subject Classification

11F33 11F37 11F46 11F55 11F85 11F03 



I am grateful to Harris, Skinner, and Urban for helpful conversations and suggestions while I was working on this paper. I am also very thankful to Hida and K.-W. Lan for answering my questions about q-expansions.


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Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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