Theta operators on Hodge type Shimura varieties

TL;DR

The paper develops a comprehensive framework for mod $p$ weight-shifting differential operators on Hodge type Shimura varieties at hyperspecial level. It introduces two operator families—basic theta operators and theta linkage maps—constructed via Verma-module methods and geometric realizations, with weight shifts governed by roots and representations; their Hecke-compatibility, Hasse-annulation behavior, and codimension-one restrictions are established. The authors connect these operators to Serre weight entailments, providing generic results for GL$_4$ and unitary groups, and non-generic examples for GL$_3$, in ways that illuminate the weight part of Serre’s conjecture and Breuil–Mezard phenomena. A de Rham realization functor is constructed to relate Verma-module data to crystalline and de Rham cohomology, enabling a de Rham–BGG perspective on cohomology and extending the integral lowest alcove BGG decomposition. Altogether, the work provides powerful tools for weight shifting, structure of automorphic vector bundles, and the interplay between representation theory and arithmetic geometry in the mod $p$ setting.

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.

Figures (2)

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Theorems & Definitions (228)

• Proposition 1.0.1
• Proposition 1.0.2
• Definition 1.1.1
• Theorem 1.1.2
• Remark 1.1.3
• Theorem 1.1.4: \ref{['Verma-linkage']}, \ref{['linkage-map-composition-basic']}
• Theorem 1.1.5: \ref{['injective-linkage']}
• Definition 1.1.6
• Theorem 1.1.7
• Remark 1.1.8