$\mathcal{L}$-invariants and local-global compatibility for the group $\mathrm{GL}_2/F$

TL;DR

This paper demonstrates that Fontaine-Mazur $ ext{L}$-invariants for certain $p$-adic Galois representations associated with quaternion Shimura curves can be detected within the completed cohomology, extending Breuil's results beyond $ ext{GL}_2/ ext{Q}$.

Contribution

It establishes the presence of $ ext{L}$-invariants in the completed cohomology for quaternion Shimura curves, generalizing prior $ ext{GL}_2/ ext{Q}$ results to totally real fields.

Findings

$ ext{L}$-invariants are realized in completed cohomology.

Generalization of Breuil's results to totally real fields.

Connection between local Galois representations and global cohomology.

Abstract

Let $F$ be a totally real number field, $\wp$ a place of $F$ above $p$. Let $\rho$ be a $2$-dimensional $p$-adic representation of $\mathrm{Gal}(\bar{F}/F)$ which appears in the \'etale cohomology of quaternion Shimura curves (thus $\rho$ is associated to Hilbert eigenforms). When the restriction $\rho_{\wp}:=\rho|_{D_{\wp}}$ at the decomposition group of $\wp$ is semi-stable non-crystalline, one can associate to $\rho_{\wp}$ the so-called Fontaine-Mazur $\mathcal{L}$-invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these $\mathcal{L}$-invariants in the completed cohomology group of quaternion Shimura curves, which generalizes some of Breuil's results in $\mathrm{GL}_2/\mathbb{Q}$-case.