Locally analytic vectors in the completed cohomology of quaternionic Shimura curves

Abstract: We use the methods introduced by Lue Pan to study the locally analytic vectors of the completed cohomology of Shimura curves associated to an indefinite quaternion algebra $D$ which is ramified at a prime number $p$. Let $D_p^{\times}$ be the group of units of $D$ at $p$. Using $p$-adic uniformization of the quaternionic Shimura curves, we compute the Hecke eigenspace of the completed cohomology with the Hecke eigenvalues associated to a classical automorphic form on another quaternion algebra $\bar D$ (switching invariants of $D$ at $p,\infty$). We present this locally analytic $D_p^\times$-representation using the de Rham complex of the Lubin-Tate tower of dimension $1$. This is analogous to the Breuil-Strauch conjecture for the group $\mathrm{GL}_2(\mathbb{Q}_p)$. We show that the locally analytic $D_p^{\times}$-representation does not detect the Hodge filtration of the local de Rham Galois representation at $p$ in the crystalline case, and also give applications for the locally analytic Jacquet--Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ and $D_p^\times$.