Gelfand-Kirillov dimension and mod p cohomology for GL2

Abstract

Let $p$ be a prime number, $F$ a totally real number field unramified at places above $p$ and $D$ a quaternion algebra of center $F$ split at places above $p$ and at no more than one infinite place. Let $v$ be a fixed place of $F$ above $p$ and $\overline{r}:\mathrm{Gal}(\overline{F}/F)\to {\mathrm{GL}}_2({\overline{\mathbb{F}}}_p)$ an irreducible modular continuous Galois representation which, at the place $v$, is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of ${\mathrm{GL}}_2(F_v)$ over ${\overline{\mathbb{F}}}_p$ associated to $\overline{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:\mathbb{Q}]$, as well as several related results.

arXiv commentFinal version, to appear in Inventiones Mathematicae

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Cited by in corpus (1)

• Can we dream of a 1-adic Langlands correspondence?