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Theta operators and a generic entailment for $GSp_4$

Martin Ortiz

TL;DR

The paper addresses weight shifting in mod $p$ automorphic forms for $GSp_4$ by constructing a broad family of theta operators on the flag Shimura variety and extending them to the Siegel threefold. A key outcome is a generic entailment: if a Hecke eigenform is modular for a lowest-alcove Serre weight, then it is modular for a shadow weight in an upper alcove, established via injectivity of a low-weight operator and cohomological comparison. Technically, the work develops a low-weight operator $\theta^1_{(k,l)}$, demonstrates injectivity on $H^0$ for $0\le k\le p-1$, and links to Breuil–Mezard cycles and Herzig's conjecture. The results extend known weight-part phenomena beyond $\mathrm{GL}_2$, with a roadmap for generalizing to Hodge-type and higher-rank cases, and provide a concrete mechanism for generic entailments in higher rank groups.

Abstract

We construct a new family of mod $p$ weight shifting differential operators on the Siegel threefold. In particular, we construct one operator which generalizes the classical theta cycle, whose weight shift allows for maps between $p$-restricted weights, and which is generically injective on global sections. As an application we produce a generic entailment of Serre weights, i.e. any Hecke eigenform which is modular for a generic Serre weight in the lowest alcove is also modular for a Serre weight in one of the upper alcoves. The entailed Serre weight corresponds to a shadow weight of the lowest alcove Serre weight, in Herzig's conjectural description of $W(\overlineρ)$.

Theta operators and a generic entailment for $GSp_4$

TL;DR

The paper addresses weight shifting in mod automorphic forms for by constructing a broad family of theta operators on the flag Shimura variety and extending them to the Siegel threefold. A key outcome is a generic entailment: if a Hecke eigenform is modular for a lowest-alcove Serre weight, then it is modular for a shadow weight in an upper alcove, established via injectivity of a low-weight operator and cohomological comparison. Technically, the work develops a low-weight operator , demonstrates injectivity on for , and links to Breuil–Mezard cycles and Herzig's conjecture. The results extend known weight-part phenomena beyond , with a roadmap for generalizing to Hodge-type and higher-rank cases, and provide a concrete mechanism for generic entailments in higher rank groups.

Abstract

We construct a new family of mod weight shifting differential operators on the Siegel threefold. In particular, we construct one operator which generalizes the classical theta cycle, whose weight shift allows for maps between -restricted weights, and which is generically injective on global sections. As an application we produce a generic entailment of Serre weights, i.e. any Hecke eigenform which is modular for a generic Serre weight in the lowest alcove is also modular for a Serre weight in one of the upper alcoves. The entailed Serre weight corresponds to a shadow weight of the lowest alcove Serre weight, in Herzig's conjectural description of .

Paper Structure

This paper contains 17 sections, 31 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Set up
  3. Weights and representation theory
  4. Hecke operators away from $p$
  5. Vanishing results for coherent cohomology
  6. Deformation theory
  7. Construction of the theta operators on the open strata
  8. The Igusa variety
  9. Construction on the open strata $U$
  10. Extension to $U_1$
  11. Extension to the flag Shimura variety
  12. Extending $\Theta_{1,2,3}$ to $\mathcal{F}l$
  13. $\pi_* \theta_{1,2,3}$ as maps on the Shimura variety.
  14. The entailment of Serre weights
  15. Generalities about the weight part of Serre's conjecture
  16. ...and 2 more sections

Key Result

Theorem 1.1

(entailment) Let $\lambda_0=(a,b) \in X^*(T)$ satisfying $a \ge b \ge 1, a+b<p-3$ and let $\mathfrak{m} \subset \mathbb{T}$ be a generic non-Eisenstein eigensystem such that $F(\lambda_0) \in W(\overline{\rho}_{\mathfrak{m}})$. Then where $\lambda_1=(p-b-3,p-a-3) \in C_1$ or $\lambda_2=(p+b-1,p-a-3) \in C_2$.

Theorems & Definitions (63)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3: Flag Shimura variety
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.8
  • Lemma 2.9
  • ...and 53 more