Article contents
${\mathcal{L}}$-INVARIANTS AND LOCAL–GLOBAL COMPATIBILITY FOR THE GROUP
$\text{GL}_{2}/F$
Published online by Cambridge University Press: 10 June 2016
Abstract
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.
MSC classification
- Type
- Research Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author 2016
References
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