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Differential operators mod p: analytic continuation and consequences

Eischen, E. E. and Flander, M. and Ghitza, A. and Mantovan, E. and McAndrew, A. (2019) Differential operators mod p: analytic continuation and consequences. . (Unpublished)

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This paper concerns certain modp 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 modp 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 modp 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:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Alternate Title:Analytic continuation of differential operators and applications to Galois representations
Additional Information: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.
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Australian Postgraduate AwardUNSPECIFIED
Albert Shimmins Writing Up AwardUNSPECIFIED
Record Number:CaltechAUTHORS:20210303-132200090
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108291
Deposited By: Tony Diaz
Deposited On:03 Mar 2021 21:48
Last Modified:03 Mar 2021 21:48

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