Multivariable ($\ddot{I}†$,${\mathcal{O}}_K^{\times }$)-modules and local-global compatibility

Abstract

Let $p$ be a prime number, $K$ a finite unramified extension of ${\mathbb{Q}}_p$ and $\mathbb{F}$ a finite extension of ${\mathbb{F}}_p$. Using perfectoid spaces we associate to any finite-dimensional continuous representation $\overline{\ddot{I}}$ of $\mathrm{Gal}(\overline{K}/K)$ over $\mathbb{F}$ an étale $(\ddot{I}†,{\mathcal{O}}_K^{\times })$-module $D_A^{\otimes }(\overline{\ddot{I}})$ over a completed localization $A$ of $\mathbb{F}[[{\mathcal{O}}_K]]$. We conjecture that one can also associate an étale $(\ddot{I}†,{\mathcal{O}}_K^{\times })$-module $D_A(\ddot{I}€)$ to any smooth representation $\ddot{I}€$ of ${\mathrm{GL}}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, and that moreover $D_A(\ddot{I}€)$ is isomorphic (up to twist) to $D_A^{\otimes }(\overline{\ddot{I}})$, where $\overline{\ddot{I}}$ is the underlying $2$-dimensional representation of $\mathrm{Gal}(\overline{K}/K)$. Using previous work of the same authors, we prove this conjecture when $\overline{\ddot{I}}$ is semi-simple and sufficiently generic.

Minor modifications after the referee report