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

Abstract

Let $p$ be a prime number and $F$ a totally real number field unramified at places above $p$. Let $\bar{r}:\mathrm{Gal}(\bar{F}/F)\to {\mathrm{GL}}_2(\overline{{\mathbb{F}}_p})$ 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 ${\mathrm{GL}}_2(F_v)$ over $\overline{{\mathbb{F}}_p}$ 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 $v$).

20 pages. arXiv admin note: text overlap with arXiv:2009.03127 by other authors

Cited by in corpus (1)

• Gelfand-Kirillov dimension and mod p cohomology for GL2