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Differential operators mod p: analytic continuation and consequences

Eischen, Ellen and Flander, Max and Ghitza, Alexandru and Mantovan, Elena and McAndrew, Angus (2021) Differential operators mod p: analytic continuation and consequences. Algebra and Number Theory, 15 (6). pp. 1469-1504. ISSN 1937-0652. doi:10.2140/ant.2021.15.1469. https://resolver.caltech.edu/CaltechAUTHORS:20210303-132200090

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Abstract

We study certain mod p differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the mod p reduction of the p-adic theta operators previously studied by some of the authors. In the characteristic 0, p-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole Shimura variety. On the other hand, our mod p operators extend (“analytically continue”, in the language of de Shalit and Goren) to the whole Shimura variety. As a consequence, motivated by their use by Edixhoven and Jochnowitz in the case of modular forms for proving the weight part of Serre’s conjecture, we discuss some effects of these operators on Galois representations. Our focus and techniques differ from those in the literature. Our intrinsic, coordinate-free approach removes difficulties that arise from working with q-expansions and works in settings where earlier techniques, which rely on explicit calculations, are not applicable. In contrast with previous constructions and analytic continuation results, these techniques work for any totally real base field, any weight, and all signatures and ranks of groups at once, recovering prior results on analytic continuation as special cases.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.2140/ant.2021.15.1469DOIArticle
https://arxiv.org/abs/1902.10911arXivDiscussion Paper
Alternate Title:Analytic continuation of differential operators and applications to Galois representations
Additional Information:© 2021 Mathematical Sciences Publishers. Received: 5 November 2019; Revised: 5 December 2020; Accepted: 5 January 2021; Published: 16 October 2021. Partially supported by NSF Grants DMS-1559609 and DMS-1751281. Supported by an Australian Postgraduate Award. Partially supported by an Australian Postgraduate Award and the Albert Shimmins Writing Up Award.
Funders:
Funding AgencyGrant Number
NSFDMS-1559609
NSFDMS-1751281
Australian Postgraduate AwardUNSPECIFIED
Albert Shimmins Writing Up AwardUNSPECIFIED
Subject Keywords:theta operators, mod p differential operators, mod p automorphic forms, analytic continuation
Issue or Number:6
Classification Code:MSC 2010: Primary: 11G18; Secondary: 11F03, 11F55, 11F80, 14G17, 14G35
DOI:10.2140/ant.2021.15.1469
Record Number:CaltechAUTHORS:20210303-132200090
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210303-132200090
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108291
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:03 Mar 2021 21:48
Last Modified:15 Mar 2022 19:07

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