Locally analytic vectors in the completed cohomology of quaternionic Shimura curves

TL;DR

This paper studies the structure of locally analytic vectors in the completed cohomology of quaternionic Shimura curves, connecting p-adic uniformization, automorphic forms, and Galois representations, with implications for the Jacquet--Langlands correspondence.

Contribution

It computes the locally analytic vectors of completed cohomology for quaternionic Shimura curves and relates them to the de Rham complex of Lubin-Tate towers, extending the Breuil-Strauch conjecture.

Findings

Computed Hecke eigenspaces using p-adic uniformization.

Connected locally analytic representations to de Rham complexes.

Showed the representation does not detect the Hodge filtration in the crystalline case.

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 $D_p^\times$.