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Entire theta operators at unramified primes

Eischen, E. and Mantovan, E. (2020) Entire theta operators at unramified primes. . (Unpublished)

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Starting with work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of p-adic and modp modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: 1) the analytic continuation at unramified primes p to the whole Shimura variety of the modp reduction of p-adic Maass--Shimura operators a priori defined only over the μ-ordinary locus, and 2) the construction of new modp theta operators that do not arise as the modp reduction of Maass-Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain significantly more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:Partially supported by NSF Grants DMS-1559609 and DMS-1751281.
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Record Number:CaltechAUTHORS:20210302-154532505
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108279
Deposited By: Tony Diaz
Deposited On:02 Mar 2021 23:54
Last Modified:02 Mar 2021 23:54

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