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Entire Theta Operators at Unramified Primes

Eischen, E. and Mantovan, Elena (2021) Entire Theta Operators at Unramified Primes. International Mathematics Research Notices . ISSN 1073-7928. doi:10.1093/imrn/rnab190. (In Press) https://resolver.caltech.edu/CaltechAUTHORS:20210302-154532505

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Abstract

Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of p-adic and mod p 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 mod p reduction of p-adic Maass–Shimura operators a priori defined only over the μ-ordinary locus, and (2) the construction of new mod p theta operators that do not arise as the mod p 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 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:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1093/imrn/rnab190DOIArticle
https://arxiv.org/abs/2002.09450arXivDiscussion Paper
Additional Information:© The Author(s) 2021. Published by Oxford University Press. Received August 14, 2020; Revised June 11, 2021; Accepted June 21, 2021. Communicated by Prof. Ana Caraiani. Our work on this project has benefitted from helpful conversations with Ehud de Shalit, Alex Ghitza, Eyal Goren, and Angus McAndrew, especially about their earlier papers on related topics. We completed key steps during our visit to the University of Lille, as well as the 2nd-named author’s visits to the University of Oregon and the University of Padua, and we are grateful to these institutions for their hospitality. We are also grateful to the referee for helpful suggestions. This work was supported by the National Science Foundation [DMS-1559609 and DMS-1751281].
Funders:
Funding AgencyGrant Number
NSFDMS-1559609
NSFDMS-1751281
DOI:10.1093/imrn/rnab190
Record Number:CaltechAUTHORS:20210302-154532505
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210302-154532505
Official Citation:E Eischen, E Mantovan, Entire Theta Operators at Unramified Primes, International Mathematics Research Notices, 2021, rnab190, https://doi.org/10.1093/imrn/rnab190
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108279
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:02 Mar 2021 23:54
Last Modified:15 Mar 2022 16:32

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