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

TL;DR

This paper introduces a new framework using perfectoid spaces to associate certain modules to Galois representations and smooth representations of GL2, proposing a conjecture linking these modules and proving it in specific cases.

Contribution

It proposes a conjectural link between Galois representations and smooth representations of GL2 via étale modules, extending the local-global compatibility in the mod p setting.

Findings

Established the conjecture for semi-simple, sufficiently generic cases.

Constructed étale $(phi,mathcal{O}_K^ imes)$-modules from Galois representations.

Proposed a new approach using perfectoid spaces for 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 $\overline{\rho}$ of ${\rm Gal}(\overline K/K)$ over $\mathbb{F}$ an \'etale $(\varphi,\mathcal{O}_K^\times)$-module $D_A^\otimes(\overline{\rho})$ over a completed localization $A$ of $\mathbb{F}[\![\mathcal{O}_K]\!]$. We conjecture that one can also associate an \'etale $(\varphi,\mathcal{O}_K^\times)$-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 $D_A^\otimes(\overline{\rho})$, where $\overline{\rho}$ is the underlying $2$-dimensional representation of ${\rm Gal}(\overline K/K)$. Using previous work of the same authors, we prove this conjecture when $\overline{\rho}$ is semi-simple and sufficiently generic.