On some mod $p$ representations of quaternion algebra over $\mathbb{Q}_p$

TL;DR

This paper investigates mod p representations of quaternion algebras over al p, showing they have Gelfand-Kirillov dimension 1 and analyzing Scholze's functor on supersingular representations, revealing new structural insights.

Contribution

It establishes the Gelfand-Kirillov dimension of certain mod p representations and proves the vanishing of Scholze's functor on supersingular representations, with detailed structure results.

Findings

Mod p cohomology representations have Gelfand-Kirillov dimension 1.

Scholze's functor vanishes on supersingular representations.

Finer structure theorems for Scholze's functor in reducible cases.

Abstract

Let $F$ be a totally real field in which $p$ is unramified and $B$ be a quaternion algebra which splits at at most one infinite place. Let $\overline{r}:\mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor-Wiles hypotheses. Assume that for some fixed place $v|p$, $B$ ramifies at $v$ and $F_v$ is isomorphic to $ \mathbb{Q}_p$ and $\overline{r}$ is generic at $v$. We prove that the admissible smooth representations of the quaternion algebra over $\mathbb{Q}_p$ coming from mod $p$ cohomology of Shimura varieties associated to $B$ have Gelfand-Kirillov dimension $1$. As an application we prove that the degree two Scholze's functor vanishes on supersingular representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. We also prove some finer structure theorem about the image of Scholze's functor in the reducible case.