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

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.

Final version, to appear in Compos. Math