Locally analytic vectors in the completed cohomology of quaternionic Shimura curves
Zhenghui Li, Benchao Su, Zhixiang Wu
TL;DR
The paper studies locally analytic vectors in the completed cohomology of quaternionic Shimura curves at a ramified prime p, building a quaternionic analogue of Breuil–Strauch via the Lubin–Tate tower.Using p-adic uniformization and Pan’s framework, it constructs de Rham-type D_p^×-representations τ(ρ_p) attached to 2–dimensional de Rham Galois representations ρ_p and proves a local-global compatibility statement for these representations inside the completed cohomology.A key feature is the short exact sequence 0→τ_p^{⊕2}→˜τ→τ_c→0 and the non-vanishing of τ_c, which persists even when the Jacquet–Langlands transfer τ_p vanishes; this exposes new phenomena in the p-adic Langlands program for D_p^×, especially when ρ_p is crystalline.The work also relates these representations to the Drinfeld/Lubin–Tate towers, establishes product- and cohomology-formulas that connect de Rham cohomology of LT spaces with automorphic forms, and connects the Scholze functor to the la-parts of completed cohomology, clarifying how local Galois data governs the GL_2(Q_p)-to-D_p^× correspondence.Overall, the results provide a detailed, geometric, and representation-theoretic framework for the p-adic Jacquet–Langlands correspondence in the quaternionic setting and yield precise local-global correspondences for de Rham Galois representations arising from global automorphic data.
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 $D_p^\times$.