Igusa Stacks in p-adic Shimura Uniformization

• Igusa stacks are v-stacks that parameterize isogeny classes of abelian varieties, p‑divisible groups, or shtukas with extra structure in Shimura varieties.
• They are constructed via v‑sheafification and a distinguished period map to classify G‑bundles on the Fargues–Fontaine curve, aiding in uniformization.
• Their framework controls the cohomology of Shimura varieties through spectral actions and supports local-global compatibility in the geometric Langlands program.

An Igusa stack is a v-stack or diamond that encodes the generic part of the $p$-adic uniformization for Shimura varieties, parametrizing isogeny classes of abelian varieties (or $p$-divisible groups or shtukas) with additional structure—together with their full level structure away from $p$ and a complete trivialization of their $p$-divisible group up to quasi-isogeny. Igusa stacks generalize classical Igusa varieties, providing a stack-theoretic, functorial, and uniquely defined object whose geometry controls the cohomology, period maps, and local-global compatibility of Shimura varieties across Hodge, PEL, abelian, and exceptional types, with critical applications to the geometric Langlands program and the realization of the local Langlands correspondence in $p$-adic and $\ell$-adic cohomology.

1. Definition and Construction

Let $(G,X)$ be a Shimura datum with reflex field $E$, and let $p$ be a prime with associated local field $E$ over which $p$0 is unramified. The Igusa stack, denoted $p$1 or more generally $p$2, is built as follows (Daniels et al., 2024, Kim, 22 Apr 2025, Schnelle, 12 Jan 2026, Partofard, 15 Dec 2025, Zhang, 2023):

• Functor of points: For a perfectoid space $p$3 over the residue field $p$4 (or $p$5), consider untilts $p$6 and points $p$7 of the integral model of the Shimura variety. Associated to $p$8 is an abelian scheme (or more generally, a $p$9-divisible group, $p$0-shtuka, or display) $p$1 up to prime-to-$p$2 isogeny.
• Morphisms: Between two such objects $p$3 and $p$4, a morphism is a quasi-isogeny over $p$5 preserving polarization, level structures, and crystalline Tate tensors, and inducing an isomorphism of associated $p$6-bundles on the Fargues–Fontaine curve.
• Sheafification: The presheaf is sheafified in the $p$7-topology to obtain $p$8, which forms a small Artin $p$9-stack (diamonds for $p$0-adic geometry).

A distinguished period map $p$1 to the stack of $p$2-bundles on the Fargues–Fontaine curve is naturally constructed, classifying the isomorphism class of the associated $p$3-bundle or local shtuka (Daniels et al., 2024, Zhang, 2023, Partofard, 15 Dec 2025).

2. Existence, Structurally Cartesian Diagrams, and Geometric Features

The main existence theorem for Hodge-type Shimura varieties (proved for large generality) states:

• Existence: For each neat level $p$4 there exists an Artin $p$5-stack $p$6 over $p$7, equipped with a $p$8-action and a period map $p$9.
• Cartesian Square: There is a cartesian diagram:

$p$0

where $p$1 is the diamond-analogue of the good reduction locus, $p$2 the associated Schubert cell in the affine Grassmannian, and $p$3 the Beauville–Laszlo uniformization (Daniels et al., 2024, Zhang, 2023, Partofard, 15 Dec 2025, Schnelle, 12 Jan 2026).

• Geometric Structure: $p$4 is $p$5-cohomologically smooth of virtual dimension zero. Over each Newton stratum $p$6 in $p$7, the fiber is the perfect Igusa variety $p$8—the perfection of a smooth affine scheme of dimension $p$9. The dualizing complex is $\ell$0.

Fiber product descriptions extend to abelian-type (Schnelle, 12 Jan 2026), PEL-type (Zhang, 2023), and exceptional (meta-unitary) Shimura varieties (Partofard, 15 Dec 2025), substantiating Scholze's conjecture on the universal role of the Igusa stack in the $\ell$1-adic uniformization and period map geometry for Shimura varieties of arbitrary type.

3. Functoriality, Uniqueness, and Axiomatic Characterization

Axiomatic frameworks for Igusa stacks have been formulated (Kim, 22 Apr 2025), specifying properties such as:

• $\ell$2-equivariance and descent data.
• Realization as moduli of $\ell$3-divisible groups (or shtukas) with $\ell$4-structure, isocrystal trivialization, and full prime-to-$\ell$5 level.
• Existence of Hodge–Tate and de Rham period maps.
• Uniformization at infinite level by universal covers (local Shimura varieties).
• Finite-level quotients are representable as perfectoid spaces or diamonds.
• Compatibility with local Shimura varieties, $\ell$6-completeness, (partial) properness, and compatibility with boundaries and compactifications.

The main uniqueness theorem: If two Igusa stacks satisfy these axioms, they are canonically isomorphic as $\ell$7-sheaves, equivariant with respect to all structures. Functoriality is automatic—any homomorphism of Shimura data induces a morphism of the associated Igusa stacks, compatible with all relevant operations. Existence of Igusa stacks descends to Shimura subdata by restriction to closed substacks corresponding to additional Hodge tensors (Kim, 22 Apr 2025).

4. Role in Cohomology and the Local Langlands Correspondence

A central structural feature is the realization of the cohomology of Shimura varieties in terms of the Igusa stack:

• Sheaf Construction: The direct image $\ell$8 is a $\ell$9-equivariant complex of sheaves on $(G,X)$0 controlling the cohomology of the associated Shimura variety $(G,X)$1.
• Hecke Operators and Spectral Action: Via the Fargues–Scholze spectral action, Hecke operators indexed by the Hodge cocharacter act on $(G,X)$2. Explicitly, restriction to the neutral component and the derived global sections $(G,X)$3 are expressed as $(G,X)$4, recovering and generalizing the Eichler–Shimura relation (Daniels et al., 2024).
• Perverse $(G,X)$5-Structure and Vanishing: When $(G,X)$6 is proper, $(G,X)$7 is ULA and perverse of weight 0 (with respect to the Newton stratification and Harder–Narasimhan index). Consequently, the cohomology $(G,X)$8 vanishes outside the middle degree for the component localized at a semisimple $(G,X)$9-parameter $E$0—a generalization of the torsion-free results of Caraiani–Scholze.
• Mantovan Product Formula: Over the $E$1-th Newton stratum, the cohomology admits a natural filtration whose graded pieces are tensor products of the cohomology of the local Igusa cover, the Rapoport–Zink (local Shimura) space, and the global group—realizing the local-global compatibility with the local Langlands program (Partofard, 15 Dec 2025).

Recent works apply unipotent categorical local Langlands theory to the derived category of sheaves on the Igusa stack, showing, after spectral localization, that the cohomology is concentrated in the middle degree and functorially realizes the expected correspondence for generic $E$2-parameters (Partofard, 15 Dec 2025).

5. Compactification, Newton Stratification, and Shtuka Theory

Igusa stacks coherently extend to integral and compactified models (Zhang, 2023, Partofard, 15 Dec 2025):

• Minimal and Toroidal Compactification: The minimal compactification of the Igusa stack is constructed via affinization and partial proper hulls, and the resulting diagram remains cartesian upon extension to the boundary, paralleling the behavior of the compactified Shimura variety.
• Shtuka-theoretic Formulation: In the function-field setting, Igusa stacks are constructed over the moduli of global $E$3-shtukas, with local data governed by isogeny classes (central leaves), and Galois symmetry implemented by pro-étale towers.
• Newton Stratification: Fibers over $E$4 correspond to admissible Newton strata, and, on each stratum, one obtains product formulas (geometric Mantovan) expressing the Shimura variety as quotients (stacks) of the product of the local Igusa variety and the Rapoport–Zink/local Shimura space by the group of self-quasi-isogenies.

6. Extensions: Abelian, Exceptional, and Motivic Igusa Stacks

The theory of Igusa stacks is established for:

• Abelian-Type Shimura Varieties: Using hybrid moduli problems admitting PEL and abelian data, Igusa stacks are constructed for types (A even) and (C) and extended to general abelian type, confirming the rational fiber product conjecture for these loci (Schnelle, 12 Jan 2026).
• Exceptional/Meta-Unitary Cases: For meta-unitary Shimura varieties, modern shtuka-stack techniques and display–gluing formalism yield Igusa stacks compatible with the full Scholze fiber-product description and allow for the realization of local-global compatibility results (Partofard, 15 Dec 2025).
• Stack-Theoretic Motivic Igusa Functions: Motivic zeta functions indexed by Igusa stacks enrich the theory by encoding orbit data for endomorphisms up to coordinate change and yield rationality properties in formal neighbourhoods of singularities (Stout, 2023).

7. Open Problems and Outlook

Ongoing and future directions include:

• Full generalization of the existence and functoriality results to all (including non-algebraic type) Shimura varieties, confirmed for a wide class but still open in maximal generality.
• Precise stack-theoretic comparison between motivic, auto-Igusa, and canonical auto-Igusa zeta functions and their poles in the context of formal deformations (Stout, 2023).
• Expansion of the categorical local Langlands correspondence in the setting of Igusa stacks, exploring the depth and extent to which perverse, unipotent, and spectral structures of the stack control the cohomology and automorphy of associated Shimura varieties.

Igusa stacks thus serve as a central organizing object unifying integral $E$5-adic geometry, Shimura uniformization, period map theory, arithmetic local Langlands, and stack-theoretic motivic integration in the modern landscape of arithmetic geometry and automorphic representation theory.