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Note on a certain category of mod $p$ representations

Reinier Sorgdrager

Abstract

Let $p>3$ be a prime number, $f\geq1$ an integer. We consider a certain full subcategory $\mathcal C$ of the category of smooth admissible mod $p$ representations of either $\text{GL}_2\mathbf Q_{p^f}$ or of the group of units of the quaternion algebra over $\mathbf Q_{p^f}$. This category was introduced in the context of the mod $p$ Langlands program by Breuil-Herzig-Hu-Morra-Schraen in the $\text{GL}_2$-case and by Hu-Wang in the quaternion case. We prove that whether a smooth admissible mod $p$ representation $π$ (with central character) belongs to $\mathcal C$ is completely determined by the restriction of $π$ to an arbitrarily small open subgroup.

Note on a certain category of mod $p$ representations

Abstract

Let be a prime number, an integer. We consider a certain full subcategory of the category of smooth admissible mod representations of either or of the group of units of the quaternion algebra over . This category was introduced in the context of the mod Langlands program by Breuil-Herzig-Hu-Morra-Schraen in the -case and by Hu-Wang in the quaternion case. We prove that whether a smooth admissible mod representation (with central character) belongs to is completely determined by the restriction of to an arbitrarily small open subgroup.

Paper Structure

This paper contains 6 sections, 9 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. $p$-valuations
  3. Ordered basis
  4. Associated graded rings and modules
  5. The annihilation condition
  6. Comparing annihilation conditions and proof of Theorem main

Key Result

Theorem 1

Let $\pi$ and $\pi'$ be smooth representations of $G_\text{big}$ ($\mathop{\mathrm{GL}}\nolimits_2K$ or $D^\times$) with same fixed central character. Suppose $\pi'\in\mathcal{C}$ and $\pi|_H\cong\pi'|_H$ for some open subgroup $H$ of $G_\text{big}$. Then $\pi\in\mathcal{C}$.

Theorems & Definitions (19)

  • Theorem 1
  • Definition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6: BHHMS1,hu2024modprepresentationsquaternion
  • Proposition 7
  • proof
  • Definition 8
  • ...and 9 more