Completed Cohomology of Shimura Curves

• Completed cohomology of Shimura curves is a p-adic refinement of classical cohomology that encodes automorphic forms and local–global correspondences via perfectoid towers.
• It integrates p-adic Hodge theory and analytic representation theory to establish vanishing, structure, and classicality theorems, with direct links to Jacquet–Langlands correspondences.
• The framework underpins eigenvariety constructions and level-lowering principles, offering deep insights into p-adic automorphic deformations and the local Langlands program.

Completed cohomology of Shimura curves provides a $p$-adic analytic refinement of the $\ell$-adic and classical cohomological structures associated to towers of arithmetic quotients attached to quaternionic or unitary algebraic groups. By systematically exploiting the rich interplay between the geometry of the infinite-level tower, $p$-adic Hodge theory, and analytic representation theory, completed cohomology encodes the totality of $p$-adic automorphic forms and their local–global correspondences—including $p$-adic and mod $p$ local Langlands, Jacquet–Langlands correspondences, and classicality phenomena. Recent advances have elucidated the geometric and analytic structure of completed cohomology for Shimura curves via comparisons with sheaf cohomology on perfectoid covers of associated flag varieties, leading to strong vanishing, structure, and classicality theorems.

1. Definitions, Foundational Constructions, and Basic Properties

Given a Shimura curve datum $(G, X)$—where $G$ is a quaternionic or unitary group such that $X$ is a one-dimensional Hermitian symmetric domain—the completed cohomology at prime $p$ is defined by first considering the tower of Shimura curves $\Sh_{K_pK^p}(G, X)$ over $\C_p$ as $K_p \subset G(\Q_p)$ varies over compact open subgroups for fixed tame level $K^p \subset G(\A_f^p)$. The completed cohomology groups are then

$\widetilde H^i(K^p, \Z_p) = \varprojlim_{s} \varinjlim_{K_p} H^i_{\text{ét}} \big( \Sh_{K^pK_p}, \Z/p^s \big),$

with possible variants for torsion or compact supports. Tensoring with $\Q_p$ and passing to locally analytic vectors yields Banach and locally analytic representations for the local group $G(\Q_p)$, with commuting actions of the absolute Galois group and the spherical Hecke algebra—see (Hansen et al., 2020, Camargo, 2022), and (Newton, 2011).

The rational vanishing theorem, first predicted by Calegari–Emerton, holds for Shimura curves: for dimension one, $\widetilde H^i = 0$ for $i > 1$ and $\widetilde H^1$ is $p$-torsion-free (Hansen et al., 2020). The only interesting completed cohomology groups are thus in degrees $0$ and $1$, with degree $1$ carrying rich automorphic and Galois-theoretic information.

2. Perfectoid Shimura Curves and Hodge–Tate Comparison

Central to the modern description is the perfectoid structure of infinite-level Shimura curves. The inverse limit $\Sh_{K^p} = \varprojlim_{K_p} \Sh_{K_pK^p}$, equipped with the Hodge–Tate period map $\pi_{\HT}:\; \Sh_{K^p} \to \Fl$ to the flag variety $\Fl=G/P_\mu$, admits a pro-étale cover $\Fl_\infty$ which is perfectoid (Aoki, 14 Aug 2025, Hansen et al., 2020). The geometric input is the construction of these perfectoid covers and period maps, ensuring that the associated Banach as well as locally analytic completed cohomology admits a cohomological description via sheaves on $\Fl_\infty$: $\big(\widetilde H^i(K^p, \Q_p) \widehat\otimes_{\Q_p} \C_p \big)^{\la} \cong H^i(\Fl_\infty, \O^{\la}_{K^p}),$ where $\O^{\la}_{K^p} \subset \pi_{\HT,*}\widehat \O_{\Sh_{K^p}}$ is the subsheaf of locally analytic sections under $G(\Q_p)$ (Aoki, 14 Aug 2025, Camargo, 2022).

On affinoid opens of $\Fl$ stable under compact subgroups, their preimages in the perfectoid Shimura curve tower are affinoid perfectoid spaces with strongly controlled cohomological and analytic properties.

3. Locally Analytic Vectors, Sen Theory, and the Hodge Criterion

The subspaces of locally analytic vectors in completed cohomology play a critical structural role. For Banach $G(\Q_p)$-modules $V$, locally analytic vectors are those whose $G(\Q_p)$-orbit maps are locally given by convergent power series; these subspaces are dense and inherit an admissible locally analytic representation structure.

The comparison between cohomology of the structure sheaf on perfectoid towers and locally analytic completed cohomology is mediated by the geometric Sen operator, which arises from interpreting analytic vector bundles and their Higgs fields pulled back from the flag variety via the Hodge–Tate period map. Explicitly, on $\Fl = \P^1$ one obtains a short exact sequence for locally analytic sheaves: $0 \to \O^{\la}_{K^p} \to \pi_{\HT,*}\widehat \O \xrightarrow{\nabla_{\HT}} \pi_{\HT,*}(\widehat\O(-1) \otimes \Omega^1(\log)) \to 0,$ and the associated spectral sequences collapse in the curve case, so that

$\widetilde H^1(K^p, \Q_p)^{\la} \cong H^1(\P^1, \O^{\la}_{K^p})\,.$

The proof employs $p$-adic functional-analytic techniques ensuring $\LA$-acyclicity for admissible Banach modules, justifying the passage to global sections (Aoki, 14 Aug 2025, Camargo, 2022).

4. Applications: Automorphic Forms, Jacquet–Langlands, and Local–Global Compatibility

Completed cohomology interpolates $p$-adic and classical automorphic forms and realizes $p$-adic functoriality principles. The $G(\Q_p)$-representation $\widetilde H^1(K^p, \Q_p)$ contains all classical forms and their $p$-adic deformations. In the modular curve case, after passage to the appropriate eigencomponent, one recovers Emerton's completed cohomology, Hida families, and overconvergent eigenforms (Hansen et al., 2020, Newton, 2011).

For quaternionic and unitary Shimura curves, the locally analytic completed cohomology groups facilitate a $p$-adic Jacquet–Langlands correspondence, relating automorphic representations between quaternionic and general linear groups, and interpolating this correspondence in families (Li et al., 20 Jan 2026, Newton, 2011). The explicit geometric realization of Hecke eigenspaces and the connection with de Rham cohomology of Lubin–Tate and Drinfeld towers—implemented at the $C$-analytic level via the Hodge–Tate period map—are critical in establishing these correspondences (Qiu et al., 15 May 2025, Li et al., 20 Jan 2026).

In the mod $p$ setting, completed cohomology realizes diagrams encoding the expected parameters for mod $p$ local Langlands, as precisely determined by the restriction of the residual Galois representation to decomposition groups at $p$. For semisimple cases, the associated Breuil functor produces $(\varphi, \Gamma)$-modules identified with tensor inductions of these local Galois representations under Fontaine's equivalence (Dotto et al., 2019).

5. Classicality, Cohomological Criteria, and Geometric Realizations

A major thread is the extraction of classicality criteria and the description of locally analytic vectors inside completed cohomology as detecting de Rham and Hodge-theoretic properties of Galois representations. For unitary and quaternionic Shimura curves, if a two-dimensional Galois representation arises in the locally $\sigma$-analytic or $\sigma$-algebraic vectors of completed cohomology, then it is $\sigma$-de Rham and arises from a classical automorphic form of the corresponding weight. The precise geometric realization, over the Drinfeld or Lubin–Tate towers, produces exact sequences identifying the cohomological classes with extensions controlled by the Hodge filtration, confirming conjectures due to Breuil and others (Qiu et al., 15 May 2025).

Moreover, these structural properties are reflected in exact functorial correspondences: the locally analytic Jacquet–Langlands transfer is constructed at the Banach representation level and identified via the local $p$-adic Langlands for $\GL_2$ with the image of completed cohomology under Scholze's "Lubin–Tate" functor (Li et al., 20 Jan 2026).

6. Eigenvarieties, Level-Lowering, and $p$-adic Families

The completed cohomology spaces serve as input for eigenvariety constructions, interpolating automorphic forms and Galois representations in rigid-analytic families. Applying the locally analytic Jacquet–Emerton functor, one obtains a coherent sheaf over the weight space whose relative spectrum is the eigencurve for the quaternionic forms (Newton, 2011). The geometric properties of completed cohomology guarantee the existence and coherence of these spaces, and enable precise statements of level-lowering principles—generalizing Mazur’s principle for modular forms to quaternionic settings (Newton, 2011).

The Hecke algebra acts faithfully on completed cohomology, and each "classical" or "overconvergent" point in the eigenvariety corresponds to a system of Hecke eigenvalues and a $p$-adic Galois representation interpolating the local–global compatibility predictions.

---

Key results on the locally analytic structure, perfectoid towers, Hodge–Tate comparison, and functoriality are obtained in (Aoki, 14 Aug 2025, Hansen et al., 2020, Camargo, 2022, Li et al., 20 Jan 2026, Qiu et al., 15 May 2025, Newton, 2011), and (Dotto et al., 2019). These works collectively provide a cohesive framework for the study of the $p$-adic, analytic, and geometric aspects of Shimura curve cohomology and their role in the $p$-adic Langlands program.