\(\mathcal{L}\)-invariants and local-global compatibility for the group \(\mathrm{GL}_2/F\). (English) Zbl 1376.11082
Author’s abstract: Let \(F\) be a totally real number field, \(\wp\) a place of \(F\) above \(p\). Let \(\rho\) be a \(2\)-dimensional \(p\)-adic representation of \(\text{Gal}(\overline{F}/F)\) which appears in the étale cohomology of quaternion Shimura curves (thus \(\rho\) is associated to Hilbert eigenforms). When the restriction \(\rho_{\wp}:=\rho\mid_{D_{\wp}}\) at the decomposition group of \(\wp\) is semistable noncrystalline, one can associate to \(\rho_{\wp}\) 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 C. Breuil’s results [in: Représentations \(p\)-adiques de groupes \(p\)-adiques III: Méthodes globales et géométriques. Paris: Société Mathématique de France. 65–115 (2010; Zbl 1246.11106)] in the \(\text{GL}_{2}/\mathbb{Q}\)-case.
Reviewer: Laurent Berger (Lyon)
MSC:
| 11S37 | Langlands-Weil conjectures, nonabelian class field theory |
| 11F80 | Galois representations |
| 11F85 | \(p\)-adic theory, local fields |
| 14F30 | \(p\)-adic cohomology, crystalline cohomology |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
Keywords:
\(p\)-adic local Langlands correspondence; completed cohomology; \(p\)-adic Hodge theory; Hilbert modular form; local-global compatibility; locally analytic representation; \(\mathcal{L}\)-invariantCitations:
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