${\mathcal{L}}$-INVARIANTS AND LOCAL–GLOBAL COMPATIBILITY FOR THE GROUP $\text{GL}_{2}/F$. (10th June 2016)

• Record Type: Journal Article
• Authors: DING, YIWEN
• Is Part Of: Forum of mathematics. Volume 4(2016)
• Publication Date: 2016-06-10
• Subjects: 11S37 (primary), -- 11S80 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2016.9
• Languages: English
• ISSNs: 2050-5094

Abstract

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.