L-INVARIANTS AND LOCAL–GLOBAL COMPATIBILITY FOR THE GROUP GL_2/F. (10th June 2016) - DING, YIWEN
Abstract : Let F be a totally real number field, a place of F above p . Let ρ be a 2 -dimensional p -adic representation of Gal(F/F) which appears in the étale cohomology of quaternion Shimura curves (thus ρ is associated to Hilbert eigenforms). When the restriction ρ_:=ρ|_D_ at the decomposition group of is semistable noncrystalline, one can associate to ρ_ the so-called Fontaine–Mazur L -invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these 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 GL_2/Q -case.