A semisimple mod $p$ Langlands correspondence in families for $G{L}_2({\mathbb{Q}}_p)$

Abstract

This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and ${\mathcal{H}}_{{\overline{\mathbb{F}}}_p}^{(1)}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $\mathbf{G}{\mathbf{L}}_2(F)$. In arXiv:2007.01364v1 we have constructed a parametrization of the ${\mathcal{H}}_{{\overline{\mathbb{F}}}_p}^{(1)}$-modules by certain $\widehat{\mathbf{G}{\mathbf{L}}_2}({\overline{\mathbb{F}}}_p)$-Satake parameters, together with an antispherical family of ${\mathcal{H}}_{{\overline{\mathbb{F}}}_p}^{(1)}$-modules. Here we let $F={\mathbb{Q}}_p$ (and $p\ge 5$) and construct a morphism from $\widehat{\mathbf{G}{\mathbf{L}}_2}({\overline{\mathbb{F}}}_p)$-Satake parameters to $\widehat{\mathbf{G}{\mathbf{L}}_2}({\overline{\mathbb{F}}}_p)$-Langlands parameters. As a result, we get a version in families of Breuil's semisimple mod $p$ Langlands correspondence for $\mathbf{G}{\mathbf{L}}_2({\mathbb{Q}}_p)$ and of Paškūnas' parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $\mathbf{G}{\mathbf{L}}_2({\mathbb{Q}}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $\widehat{\mathbf{G}{\mathbf{L}}_2}({\overline{\mathbb{F}}}_p)$-representations of the Galois group $\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$.

23 pages