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

Let $F_{\mathrm{\wp }}$ 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 $\mathrm{G}\mathrm{a}\mathrm{l}(\overline{{\mathbb{Q}}_p}\mathrm{/}{F}_{\mathrm{\wp }})$-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.