#### Vol. 15, No. 6, 2021

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Differential operators mod $p$: analytic continuation and consequences

### Ellen Eischen, Max Flander, Alexandru Ghitza, Elena Mantovan and Angus McAndrew

Vol. 15 (2021), No. 6, 1469–1504
##### 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.

##### Keywords
theta operators, mod $p$ differential operators, mod $p$ automorphic forms, analytic continuation
##### Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 11F03, 11F55, 11F80, 14G17, 14G35
##### Milestones
Revised: 5 December 2020
Accepted: 5 January 2021
Published: 16 October 2021
##### Authors
 Ellen Eischen Department of Mathematics University of Oregon Eugene, OR United States Max Flander PlanGrid San Francisco, CA United States Alexandru Ghitza School of Mathematics and Statistics University of Melbourne Parkville, VIC Australia Elena Mantovan Department of Mathematics California Institute of Technology Pasadena, CA United States Angus McAndrew Department of Mathematics and Statistics Boston University Boston, MA United States