Multivariable $(\varphi, \mathcal{O}_K^\times)$-modules and local-global compatibility
Résumé
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 $\rho$ of $Gal(\overline{K}/K)$ over $\mathbb{F}$ an étale $(\varphi, \mathcal{O}_K^\times)$-module $D^{\otimes}_A(\rho)$ over a completed localization $A$ of $\mathbb{F}[[ \mathcal{O}_K]]$. We conjecture that one can also associate an étale $(\varphi, \mathcal{O}_K^\times)$-module $D_A(\pi)$ to any smooth representation $\pi$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, and that moreover $D_A (\pi)$ is isomorphic (up to twist) to $D^{\otimes}_A (\rho)$, where $\rho$ is the underlying 2-dimensional representation of $Gal(\overline{K}/K)$. Using previous work of the same authors, we prove this conjecture when $\rho$ is semi-simple and sufficiently generic.
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