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Differential operators and families of automorphic forms on unitary groups of arbitrary signature

Eischen, Ellen and Fintzen, Jessica and Mantovan, Elena and Varma, Ila (2018) Differential operators and families of automorphic forms on unitary groups of arbitrary signature. Documenta Mathematica, 23 . pp. 445-495. ISSN 1431-0635. doi:10.25537/dm.2018v23.445-495.

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In the 1970’s, Serre exploited congruences between qexpansion coefficients of Eisenstein series to produce p-adic families of Eisenstein series and, in turn, p-adic zeta functions. Partly through integration with more recent machinery, including Katz’s approach to p-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz’s and Serre’s ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q-expansions when the signature is of the form (a, b), a ≠ b. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre–Tate expansions, we explicitly describe the action of differential operators on the Serre–Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p-adic measure taking values in the space of p-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p-adic family of Eisenstein series. One application of our results is to the recently completed construction of p-adic L-functions for unitary groups by the first named author, Harris, Li, and Skinner.

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Additional Information:© 2018 FIZ Karlsruhe GmbH. Attribution 4.0 International (CC BY 4.0). Received: January 1, 2017; Revised: August 31, 2017. We would like to thank Ana Caraiani very much for contributing to initial conversations about topics in this paper. We would also like to thank the referee for a careful reading and helpful suggestions. E.E.’s research was partially supported by NSF Grants DMS-1249384 and DMS-1559609. J.F.’s research was partially supported by the Studienstiftung des deutschen Volkes. E.M.’s research was partially supported by NSF Grant DMS-1001077. I.V.’s research was partially supported by a National Defense Science and Engineering Fellowship and NSF Grant DMS-1502834.
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Studienstiftung des deutschen VolkesUNSPECIFIED
National Defense Science and Engineering Graduate (NDSEG) FellowshipUNSPECIFIED
Subject Keywords:p-adic automorphic forms, differential operators, Maass operators, Shimura varieties
Classification Code:MSC: 14G35, 11G10, 11F03, 11F55, 11F60
Record Number:CaltechAUTHORS:20170712-123707950
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79012
Deposited By: Tony Diaz
Deposited On:12 Jul 2017 21:15
Last Modified:15 Nov 2021 17:44

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