# Differential operators mod $p$: analytic continuation and consequences

## TL;DR

This paper introduces mod p differential operators on automorphic forms over Shimura varieties, extending their analytic continuation beyond the ordinary locus and exploring implications for Galois representations, with a novel coordinate-free approach.

## Contribution

It develops a new intrinsic, coordinate-free method for mod p differential operators that extends their analytic continuation to entire Shimura varieties, applicable in broad settings.

## Key findings

- Operators agree with p-adic theta operators on the ordinary locus
- Operators extend analytically to the whole Shimura variety
- Implications for Galois representations and Serre's conjecture

## Abstract

This paper concerns 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.