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

Abstract

Let $F$ be a totally real number field, $\wp$ a place of $F$ above $p$. Let $ρ$ be a $2$-dimensional $p$-adic representation of $\mathrm{Gal}(\bar{F}/F)$ which appears in the étale cohomology of quaternion Shimura curves (thus $ρ$ is associated to Hilbert eigenforms). When the restriction $ρ_{\wp}:=ρ|_{D_{\wp}}$ at the decomposition group of $\wp$ is semi-stable non-crystalline, one can associate to $ρ_{\wp}$ 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.