On the mod $p$ cohomology for $\operatorname{GL}_2$

TL;DR

This paper investigates the structure of mod p cohomology for GL_2 over totally real fields, demonstrating that many associated smooth representations have maximal Gelfand--Kirillov dimension, extending previous results.

Contribution

It provides a unified proof that many admissible smooth representations associated to a modular Galois representation have maximal Gelfand--Kirillov dimension, regardless of semisimplicity.

Findings

Many representations have Gelfand--Kirillov dimension equal to [F_v:Q_p]

Extends previous work to all cases, including non-semisimple representations

Provides a unified proof for the structure of mod p cohomology representations

Abstract

Let $p$ be a prime number and $F$ a totally real number field unramified at places above $p$. Let $\bar{r}:\operatorname{Gal}(\bar F/F)\rightarrow\operatorname{GL}_2(\bar{\mathbb{F}_p})$ be a modular Galois representation which satisfies the Taylor-Wiles hypothesis and some technical genericity assumptions. For $v$ a fixed place of $F$ above $p$, we prove that many of the admissible smooth representations of $\operatorname{GL}_2(F_v)$ over $\bar{\mathbb{F}_p}$ associated to $\bar{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:\mathbb{Q}_p]$. This builds on and extends the work of Breuil-Herzig-Hu-Morra-Schraen and Hu-Wang, giving a unified proof in all cases ($\bar{r}$ either semisimple or not at $v$).