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${\mathcal{L}}$ -INVARIANTS AND LOCAL–GLOBAL COMPATIBILITY FOR THE GROUP $\text{GL}_{2}/F$

Part of: Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  10 June 2016

YIWEN DING
YIWEN DING*
Affiliation:
Department of Mathematics, Imperial College London, UK; [email protected]

Abstract

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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

Primary: 11S37: Langlands-Weil conjectures, nonabelian class field theory
Secondary: 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e13
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Creative Commons
Creative Common License - CCCreative Common License - BY
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

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