Theta operators on Hodge type Shimura varieties

TL;DR

This paper introduces new mod p weight shifting differential operators called theta operators on Hodge type Shimura varieties, generalizing classical operators and relating to Serre's conjecture, with applications to weight conjectures for specific groups.

Contribution

It constructs a new family of theta operators and linkage maps on Hodge type Shimura varieties, extending classical concepts and connecting to Serre weight conjectures.

Findings

Constructed basic theta operators for Hodge type Shimura varieties.

Developed theta linkage maps linking automorphic vector bundles.

Proved injectivity properties and provided examples related to Serre weights.

Abstract

We construct a new family of mod $p$ weight shifting differential operators on Hodge type Shimura varieties at hyperspecial level. First we construct basic theta operators, labelled by positive roots, that generalize Katz's theta operator for modular forms. Secondly we construct theta linkage maps, these are operators between automorphic vector bundles with linked weights, which can be thought of as generalizations of the classical theta cycle of Tate--Jochnowitz. In particular, there exist such maps within the $p$-restricted region, whose weight shifts are directly related to the conjectures of Herzig on the weight part of Serre's conjecture. We explain the relation between the two operators, and we prove some properties about them, e.g., the injectivity of some of them in a generic locus of the $p$-restricted region. As an application, we produce an example of a generic entailment of Serre weights for the groups $GL_{4,\mathbb{Q}_p}$ and $U(4)_{\mathbb{Q}_p}$, by combining the method of arxiv:2410.09602 with our stronger results about theta operators.