Published online by Cambridge University Press: 10 June 2016
Let $F$ be a totally real number field,
${\wp}$ a place of
$F$ above
$p$ . Let
${\it\rho}$ be a
$2$ -dimensional
$p$ -adic representation of
$\text{Gal}(\overline{F}/F)$ which appears in the étale cohomology of quaternion Shimura curves (thus
${\it\rho}$ is associated to Hilbert eigenforms). When the restriction
${\it\rho}_{{\wp}}:={\it\rho}|_{D_{{\wp}}}$ at the decomposition group of
${\wp}$ is semistable noncrystalline, one can associate to
${\it\rho}_{{\wp}}$ 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 [Breuil, Astérisque, 331 (2010), 65–115] in the
$\text{GL}_{2}/\mathbb{Q}$ -case.