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p-adic families of automorphic forms in the µ-ordinary setting

Eischen, E. and Mantovan, E. (2021) p-adic families of automorphic forms in the µ-ordinary setting. American Journal of Mathematics, 143 (1). pp. 1-52. ISSN 0002-9327.

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We develop a theory of p-adic automorphic forms on unitary groups that allows p-adic interpolation in families and holds for all primes p that do not ramify in the reflex field E of the associated unitary Shimura variety. If the ordinary locus is nonempty (a condition only met if p splits completely in E), we recover Hida's theory of p-adic automorphic forms, which is defined over the ordinary locus. More generally, we work over the µ-ordinary locus, which is open and dense. By eliminating the splitting condition on p, our framework should allow many results employing Hida's theory to extend to infinitely many more primes. We also provide a construction of p-adic families of automorphic forms that uses differential operators constructed in the paper. Our approach is to adapt the methods of Hida and Katz to the more general µ-ordinary setting, while also building on papers of each author. Along the way, we encounter some unexpected challenges and subtleties that do not arise in the ordinary setting.

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Additional Information:© 2021 Johns Hopkins University Press. Manuscript received January 19, 2018; revised January 28, 2019. Research of the first author supported in part by NSF grant DMS-1559609 and NSF CAREER Grant DMS-1751281. We thank H. Hida for feedback on the initial idea for this project. We are also very grateful to E. de Shalit and E. Goren for helpful feedback on the first version of this paper, which led to key improvements. Our work on this paper benefitted from conversations with several additional mathematicians concerning topics closely related to this paper. E.M. thanks B. Moonen for a discussion about his earlier work on which portions of this paper rely significantly, and she thanks M.-H. Nicole for a discussion about his work with W. Goldring on the µ-ordinary Hasse invariant. We also thank E. Rains for alerting us to the Littlewood–Richardson rule mentioned in Section 2.4.1, and we thank M. Harris for helpful feedback about references. We are grateful to J. Fintzen and I. Varma for conversations concerning the collaboration [EFMV18], whose influence is seen here. We thank Caltech and the University of Oregon for hosting us during this collaboration.
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Official Citation:Eischen, E., & Mantovan, E. (2021). p-adic families of automorphic forms in the µ-ordinary setting. American Journal of Mathematics 143(1), 1-52. doi:10.1353/ajm.2021.0006.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108043
Deposited By: George Porter
Deposited On:16 Feb 2021 15:28
Last Modified:16 Feb 2021 15:29

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