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A theta operator for the group $\mathrm{GSp}_4$

Leonardo Fiore

Abstract

We construct a differential operator on sheaves of $p$-adic modular forms defined over the locus of $p$-rank $\ge 1$ of the Siegel threefold, by applying a revisited version of the approach that Sean Howe recently introduced in his paper "A unipotent circle action on $p$-adic modular forms" (2020, Trans. Am. Math. Soc.) to construct the theta operator in the elliptic case.

A theta operator for the group $\mathrm{GSp}_4$

Abstract

We construct a differential operator on sheaves of -adic modular forms defined over the locus of -rank of the Siegel threefold, by applying a revisited version of the approach that Sean Howe recently introduced in his paper "A unipotent circle action on -adic modular forms" (2020, Trans. Am. Math. Soc.) to construct the theta operator in the elliptic case.
Paper Structure (27 sections, 25 theorems, 5 equations)

This paper contains 27 sections, 25 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Theta operators
  3. The Siegel threefold
  4. Our main results
  5. Outline
  6. Acknowledgments
  7. Preliminaries and conventions
  8. Actions
  9. Vector bundles
  10. Torsors and representations
  11. $p$-divisible groups
  12. The Iwasawa algebra of a profinite group
  13. Actions by $\mu_{p^\infty}$
  14. The $\mathrm{GSp}_4$ setting
  15. The Siegel threefold
  16. ...and 12 more sections

Key Result

Theorem 1

Let $f$ be a normalized cuspidal eigenform of weight $k\in \mathbb{Z}/(p-1)\mathbb{Z}$, level $\Gamma_1(N)$ and nebentypus $\varepsilon: (\mathbb{Z}/N\mathbb{Z})^\times \to E^\times$, with coefficients in some fine extension $E/\mathbb{F}_p$. Let $\rho_f: G_\mathbb{Q}\to \mathop{\mathrm{GL}}\nolimit

Theorems & Definitions (52)

  • Theorem
  • Theorem : A
  • Theorem : B
  • Proposition 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • ...and 42 more