Gelfand--Kirillov dimension and mod $p$ cohomology for inner forms of $\mathrm{GL}_2$

TL;DR

This paper computes the Gelfand--Kirillov dimension of Hecke eigenspaces in mod p cohomology for inner forms of GL_2 over totally real fields, extending previous results and providing simplified proofs in certain cases.

Contribution

It introduces a method to compute GK-dimensions for inner forms of GL_2 in mod p cohomology, including division algebra cases, and simplifies existing proofs.

Findings

Computed GK-dimension for Hecke eigenspaces in mod p cohomology.

Extended results to cases where D is a division algebra at p.

Provided a simplified proof of a theorem by Breuil--Herzig--Hu--Morra--Schraen.

Abstract

Under standard assumptions, we compute the GK-dimension of Hecke eigenspaces in the mod $p$ cohomology of an inner form $D^\times$ of $\mathrm{GL}_2$ over a totally real field unramified at $p$, allowing $D$ to be a division algebra at $p$. Our arguments also apply when $D$ is a matrix algebra at $p$, in which case they give a simplified proof of a theorem of Breuil--Herzig--Hu--Morra--Schraen.