Generalised theta operators on unitary Shimura varieties

TL;DR

This paper constructs new weight shifting operators on unitary Shimura varieties' special fibers, which could advance understanding of modular forms mod p and Serre's conjecture.

Contribution

It introduces a novel class of theta-like operators acting on EO strata of unitary Shimura varieties, extending previous work and enabling new applications.

Findings

Operators act on graded sheaves derived from EO strata

Operators are defined on lower EO strata at a good prime p

Potential applications to Hecke eigensystems and Serre's conjecture

Abstract

The main result of this paper is the construction of a new class of weight shifting operators, similar to the theta operators of arXiv:1902.10911, arXiv:1712.06969 and others, which are defined on the lower Ekedahl-Oort strata of the geometric special fibre of unitary Shimura varieties of signature $(n-1, 1)$ at a good prime $p$, split in the in the reflex field $E$, which we assume to be quadratic imaginary. These operators act on certain graded sheaves which are obtained from the arithmetic structure of the EO strata, in particular the $p$-rank on each stratum. We expect these operators to have applications to the study of Hecke-eigensystems of modular forms modulo $p$ and generalisations of the weight part of Serre's conjecture.