$\mathcal{L}$-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 $\rho$ be a $2$-dimensional $p$-adic representation of $\mathrm{Gal}(\bar{F}/F)$ which appears in the étale 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.

31 pages, comments welcome

References in corpus (1)

• On some partially de Rham Galois representations

Cited by in corpus (1)

• $\mathcal{L}$-invariants, partially de Rham families and local-global compatibility