Pro-$p$ Iwahori-Hecke modules in semisimple rank one and singularity categories

Abstract

Let $\mathfrak{F}$ be a non-archimedean local field of residue characteristic $p$ and $G$ be one of the groups ${\mathrm{GL}}_2(\mathfrak{F})$, ${\mathrm{SL}}_2(\mathfrak{F})$ or ${\mathrm{PGL}}_2(\mathfrak{F})$. Let ${\mathcal{H}}_G$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over ${\overline{\mathbb{F}}}_p$. We study the homotopy category $\mathrm{Ho}({\mathcal{H}}_G)$ of Hovey's Gorenstein projective model structure on the category of ${\mathcal{H}}_G$-modules and relate it to the singularity category $\mathrm{Sing}(X_{q,\mathbf{G}})$ of an explicit scheme. When $G={\mathrm{GL}}_2(\mathfrak{F})$, this scheme was first introduced by Dotto-Emerton-Gee \cite{DEG22}. We obtain in that case an equivalence $\mathrm{Ho}({\mathcal{H}}_{{\mathrm{GL}}_2})\simeq \mathrm{Sing}(X_{q,{\mathrm{GL}}_2})$ and recover from this Grosse-Klönne's mod-$p$ Langlands correspondence for Hecke modules \cite{GK20}, building on work of Pépin-Schmidt \cite{PeSch25_2}. We furthermore describe $\mathrm{Ho}({\mathcal{H}}_G)$ completely explicitly when $G={\mathrm{SL}}_2(\mathfrak{F})$ or ${\mathrm{PGL}}_2(\mathfrak{F})$, and make additional computations in the ${\mathrm{GL}}_2$ case.

arXiv comment31 pages; minor revisons, some references added; comments welcome