Locally analytic vectors in the completed cohomology of unitary Shimura curves

Abstract: We use the methods introduced by Lue Pan to study the locally analytic vectors in the completed cohomology of unitary Shimura curves. As an application, we prove a classicality result on two-dimensional regular $\sigma$-de Rham representations of $\text{Gal}(\bar L/L)$ appearing in the locally $\sigma$-analytic vectors of the completed cohomology, where $L$ is a finite extension of $\mathbb{Q}_p$ and $\sigma:L\hookrightarrow E$ is an embedding of $L$ into a sufficiently large finite extension $E$ of $\mathbb{Q}_p$. We also prove that if a two-dimensional representation of $\text{Gal}(\bar L/L)$ appears in the locally $\sigma$-algebraic vectors of the completed cohomology then it is $\sigma$-de Rham. Finally, we give a geometric realization of some locally $\sigma$-analytic representations of $\mathrm{GL}_2(L)$. This realization has some applications to the $p$-adic local Langlands program, including a locality theorem for Galois representations arising from classical automorphic forms, an admissibility result for coherent cohomology of Drinfeld curves, and some special cases of the Breuil's locally analytic Ext$^1$-conjecture for $\mathrm{GL}_2(L)$.