$\mathcal{L}$-invariants, partially de Rham families and local-global compatibility

Abstract

Let $F_{\wp }$ be a finite extension of ${\mathbb{Q}}_p$. By considering partially de Rham families, we establish a Colmez-Greenberg-Stevens formula (on Fontaine-Mazur $\mathcal{L}$-invariants) for (general) $2$-dimensional semi-stable non-crystalline $\mathrm{Gal}(\overline{{\mathbb{Q}}_p}/F_{\wp })$-representations. As an application, we prove local-global compatibility results for completed cohomology of quaternion Shimura curves, and in particular the equality of Fontaine-Mazur $\mathcal{L}$-invariants and Breuil's $\mathcal{L}$-invariants, in critical case.

37 pages, comments welcome

References in corpus (1)

• $\mathcal{L}$-invariants and local-global compatibility for the group ${\mathrm{GL}}_2/F$

Cited by in corpus (1)

• Companion points and locally analytic socle for ${\mathrm{GL}}_2(L)$