[論文レビュー] Locally analytic vectors in the completed cohomology of quaternionic Shimura curves

Zhenghui Li, Buchin Su+1 more|arXiv (Cornell University)|Jan 20, 2026

Advanced Algebra and Geometry被引用数 0

ひとこと要約

要約: The paper describes the locally analytic D_p^×-representations attached to de Rham Galois representations appearing in the completed cohomology of quaternionic Shimura curves, via Lubin–Tate towers and p-adic uniformization, establishing a quaternionic analogue of Breuil–Strauch for GL2(Q_p).

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 $D_p^{ imes}$ 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 $D_p^ imes$-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 $D_p^{ imes}$-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 $D_p^ imes$.

研究の動機と目的

• Motivate and construct the locally analytic D_p^×-representations associated to de Rham Galois representations with global origin.
• Describe the structure of completed cohomology via p-adic uniformization and Lubin–Tate spaces.
• Establish local-global compatibility for the D_p^×-representation τ(ρ_p) arising from de Rham ρ_p.
• Explore how τ(ρ_p) relates to Jacquet–Langlands correspondences for GL2(Q_p) and D_p^×.
• Provide non-vanishing results and structural properties (admissibility, GK-dimension) of the constructed representations.

提案手法

• Use Pan’s framework to study locally analytic vectors in completed cohomology of quaternionic Shimura curves.
• Apply p-adic uniformization to relate Shimura curve cohomology to Lubin–Tate and Drinfeld towers.
• Construct τ(ρ_p) from the de Rham complex of Lubin–Tate spaces via a short exact sequence 0→τ_p^{⊕2}→˜τ→τ_c→0 and the quotient τ(ρ_p).
• Prove a product formula tying the Lubin–Tate cohomology to automorphic forms on the definite quaternion algebra and a Shimura set.
• Demonstrate a local-global compatibility result: 𝐏𝐢𝐙̌ː(ρ)^{la}≅τ(ρ_p)^{⊕m} and analyze when ρ appears in completed cohomology.
• Discuss consequences for Scholze’s functor and p-adic Jacquet–Langlands correspondences.

実験結果

リサーチクエスチョン

• RQ1 Can one describe the locally analytic D_p^×-representation attached to a de Rham ρ_p of Gal(Q̄_p/Q_p) with HT weights 0,1 in terms of Lubin–Tate cohomology?
• RQ2 Does the completed cohomology’s locally analytic part realize a quaternionic analogue of Breuil–Strauch via τ(ρ_p)?
• RQ3 Under what global conditions does a Galois representation ρ appear in the completed cohomology, and how does τ(ρ_p) behave when τ_p=0?
• RQ4 What is the precise local-global compatibility between τ(ρ_p) and the p-adic Langlands correspondence for D_p^×?
• RQ5 How does the Lubin–Tate description interact with Scholze’s Jacquet–Langlands functor in relating GL2(Q_p) and D_p^× representations?

主な発見

• There is a D_p^×-equivariant, topological isomorphism between the la-part of the completed cohomology and τ(ρ_p)^{⊕m}.
• The locally analytic D_p^×-representation τ(ρ_p) fits into a short exact sequence 0→τ_p→τ(ρ_p)→τ_c→0, showing de Rham data is captured by a quotient involving Lubin–Tate and Jacquet–Langlands data.
• The Lubin–Tate-based representations τ_c and ˜τ are infinite‑dimensional, non-zero, and relate to the Drinfeld/Lubin–Tate towers via a product formula with automorphic forms on the definite quaternion algebra.
• τ(ρ_p) can be defined over E and is an admissible, infinite-dimensional locally analytic D_p^×-representation; its structure reflects both GL2(Q_p) and D_p^× Langlands aspects.
• τ(ρ_p) is non-vanishing even when the Jacquet–Langlands transfer τ_p vanishes, illustrating non-trivial p-adic Jacquet–Langlands phenomena.
• When ρ_p is crystalline with τ_p=0, τ(ρ_p) is still determined by the Weil–Deligne representation r_p and can be independent of the Hodge filtration, revealing new phenomena in p-adic Langlands for D_p^×.