Gelfand--Kirillov dimension and mod $p$ cohomology for $\mathrm{GL}_2$ - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Inventiones Mathematicae Année : 2023

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

Résumé

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 $\overline{r} : {\rm Gal}(\overline F/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ 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.

Dates et versions

hal-02948686 , version 1 (24-09-2020)

Identifiants

Citer

Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, Benjamin Schraen. Gelfand--Kirillov dimension and mod $p$ cohomology for $\mathrm{GL}_2$. Inventiones Mathematicae, 2023, 234 (1), pp.1-128. ⟨10.1007/s00222-023-01202-8⟩. ⟨hal-02948686⟩

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