Rapoport-Zink Space: Moduli & Stratification

• Rapoport-Zink space is a formal moduli space for deformations (up to quasi-isogeny) of p-divisible groups with extra structures, crucial for p-adic geometry and Shimura varieties.
• Its flatness and well-defined local model properties ensure that the reduction modulo p accurately reflects deformation-theoretic and group-theoretic structures.
• A group-theoretic framework via affine Deligne–Lusztig varieties and extended Weyl groups provides explicit stratifications and universal homeomorphisms to Deligne–Lusztig varieties.

A Rapoport-Zink space is a formal moduli space parameterizing deformations (up to quasi-isogeny) of a fixed $p$-divisible group with additional structure (such as endomorphisms, polarizations, or level structure), closely related to local (and global) Shimura varieties. These spaces play a pivotal role in $p$-adic geometry, the theory of moduli of abelian varieties, and are central to local and global aspects of the Langlands program, as they govern the geometry and cohomology of the supersingular or basic loci in integral models of Shimura varieties. Their geometry is intricately linked to affine Deligne–Lusztig varieties, Deligne–Lusztig theory for finite groups of Lie type, and the stratification of their special fibers via Bruhat–Tits theory. The paper of Rapoport–Zink spaces encompasses their flatness and local model properties, the explicit structure of their special fibers, as well as group-theoretic reduction techniques à la Deligne–Lusztig and the structure of the so-called admissible set.

1. Moduli Problem and Uniformization of Basic Loci

A Rapoport-Zink space $\mathcal{N}$ is constructed to parameterize deformations (up to quasi-isogeny) of a fixed $p$-divisible group $\mathbb{X}$, possibly endowed with additional endomorphism and polarization data, over complete local rings with nilpotent $p$. For PEL-type unitary Shimura varieties—particularly for groups like $\mathrm{GU}(2,4)$—$\mathcal{N}$ precisely governs the moduli of $p$-divisible groups with Hermitian (unitary) structure, matching a "framing object" $\mathbb{X}$ of signature $(2,4)$ up to quasi-isogeny and prescribed level structure.

The supersingular locus (basic Newton stratum) of a Shimura variety $\mathrm{Sh}_{\mathrm{GU}(2,4)}$ at a ramified prime is uniformized (in the sense of a canonical formal isomorphism) by the corresponding Rapoport-Zink space $\mathcal{N}$. More concretely, this locus is described by moduli functors that are interpretable via ADLVs—affine Deligne–Lusztig varieties: $$X_w(b) = \{ gI \in G(\breve{F})/I \mid g^{-1}b\sigma(g) \in IwI \},$$ where $b$ is a basic element corresponding to $\mathbb{X}$, $I$ is an Iwahori (or deeper) level parahoric, and $w$ is an element of the extended affine Weyl group. This analytic uniformization is a critical tool for transferring the local structure of Shimura varieties to explicit moduli problems involving $p$-divisible groups.

2. Flatness and Local Model Properties

A foundational property for Rapoport–Zink spaces is flatness over the base, typically the ring of integers $O_E$ of a $p$-adic field (possibly ramified). Flatness ensures that the reduction modulo $p$ of the space accurately reflects the generic fiber and that the dimension does not jump on specialization. For the GU$(2,4)$ Rapoport-Zink space at a ramified prime ($n=6$, signature (2,4)), flatness is established via a commutative algebra computation: one constructs a $6\times 6$ matrix of variables $X$, and considers the ideal

$$J(2,6) = \left\langle \operatorname{tr}(X), X^2,\, \text{all } 3\times 3\text{ minors of } X \right\rangle$$

(corresponding to the constraints of trace, nilpotency, and isotropy in the moduli problem). By exhibiting that $J(2,6)$ is radical using Gröbner basis computation, the flatness of the corresponding moduli space is rigorously established (Trentin, 2023).

This property is crucial: it implies that the geometric special fiber records, with fidelity, the intended deformation-theoretic and group-theoretic structure, and that no "unexpected" components or multiplicities pollute the special fiber.

3. Irreducible Components and Universal Homeomorphisms with Deligne–Lusztig Varieties

The structure of the reduction modulo $p$ (i.e., the special fiber) of a Rapoport–Zink space, especially for GU$(2,4)$, is meticulously described in terms of explicit families of irreducible components, each universally homeomorphic to Deligne–Lusztig (DL) varieties, either classically associated to symplectic or orthogonal groups or as generalized for this context:

• Split Hermitian Case: Two families of irreducible components emerge.
  1. For each vertex lattice $\mathcal{L}$ of maximal type (type 6), an irreducible component $N_\mathcal{L}$ is universally homeomorphic to a generalized Deligne–Lusztig variety $S_V$ for $\mathrm{Sp}_6$, typically 5-dimensional. The stratification $$S_V = \bigsqcup_{\mathcal{L}' \subset \mathcal{L}} N_{\mathcal{L}'}^\circ$$ further describes open dense pieces corresponding to deeper vertex lattice inclusions.
  2. For each $2$-modular lattice $\Lambda$ (satisfying $\Lambda^\vee = \pi^2 \Lambda$), another component $N_\Lambda^{\leq 1}$ arises, universally homeomorphic via a morphism $f: N_\Lambda \rightarrow S_{V_\pi}$, where $S_{V_\pi}$ is a closed subscheme of the Grassmannian capturing the associated semi-linear algebra data. An open dense subset $N_\Lambda^{(1)}$ is a line bundle over a generalized DL variety $R_W$ (here, typically for $\mathrm{SO}_6$).

• Non-split Hermitian Case: All irreducible components are associated to $2$-modular lattices. The universal homeomorphism property persists, but now the fibers include components which are closures of higher rank vector bundles over classical DL varieties for $\mathrm{SO}_6$, in addition to the aforementioned line bundles over generalized DL varieties. The special fiber becomes pure of dimension 4.

Key structural morphisms include: $$f: N_\mathcal{L} \to S_V, \qquad f(X, \rho, \lambda, \iota) = \ker\big(D(\rho_{X, \mathcal{L}^-})\big),$$ where $\mathcal{L}^- = \mathcal{L}^\vee$, and for $2$-modular lattices,

$$f: N_\Lambda \to S_{V_\pi}, \quad V = \Lambda^+/\Lambda^-.$$

This explicit identification enables transfer of dimension and irreducibility properties from the well-studied theory of DL varieties to the complex geometric world of $p$-adic Rapoport–Zink spaces.

4. Group-Theoretic Framework: Extended Affine Weyl Groups and Reduction Method

The local and global structure of Rapoport–Zink spaces is intricately connected to the theory of extended affine Weyl groups $\widetilde{W} = W_a \rtimes \Omega$, where $W_a$ is the affine Weyl group (type BC$_3$ in the GU$(2,4)$ setting), and $\Omega$ is a length-zero finite subgroup (order 2 in the split form).

The Bruhat–Tits stratification and the geometry of Rapoport–Zink spaces are organized via the theory of affine Deligne–Lusztig varieties (ADLVs): $$X_w(b) = \{ gI \in G(\breve{F})/I \mid g^{-1}b\sigma(g) \in IwI \},$$ where the admissible set is defined as

$$\operatorname{Adm}(\mu^\vee) = \{ w \in \widetilde{W} \mid w \leq t^{x(\mu^\vee)} \text{ for some } x \in W_0 \}.$$

Group-theoretic reduction methods à la Deligne–Lusztig enable the direct analysis of ADLVs: when certain combinatorial length conditions (e.g., $\ell(sw\sigma(s)) = \ell(w)-2$ for a simple reflection $s$) are satisfied, one obtains explicit decomposition of $X_w(b)$ into a union of locally trivial $\mathbb{G}_m$- or $\mathbb{A}^1$-bundles over simpler ADLVs. Iteration reduces arbitrary ADLVs to those for minimal length elements (often "straight elements" with respect to $\sigma$-conjugacy), which are directly identified with DL varieties for symplectic or orthogonal groups.

These group-theoretic structures underpin both the explicit description of the components and the incidences among them, linking the arithmetic geometry of Rapoport–Zink spaces to the combinatorics of Weyl groups and the geometry of flag varieties.

5. Stratification, Incidence, and Deligne–Lusztig Correspondence

The special fiber of the GU$(2,4)$ Rapoport–Zink space receives a stratification according to lattice-theoretic and group-theoretic data:

• The stratification corresponds to vertex lattices (maximal type, $2$-modular) and their inclusions, and is further refined by the identification with orbits under the group $J$ of self-quasi-isogenies.
• For vertex lattice components, the strata correspond to generalized DL varieties for $\mathrm{Sp}_6$, stratified further by inclusions (open dense subsets corresponding to inclusions $\mathcal{L}' \subset \mathcal{L}$).
• For $2$-modular lattice components, each corresponds (via universal homeomorphism) either to a DL variety for $\mathrm{SO}_6$, or in some non-split cases, to the closure of a rank-2 vector bundle over a classical DL variety.

This highly structured stratification admits explicit dimension formulas and incidence relations, reflecting both arithmetic and representation-theoretic invariants intrinsic to the moduli problem.

6. Significance and Applications

The explicit description of the special fiber in terms of Deligne–Lusztig varieties, stratification via vertex lattices, and the group-theoretic construction via ADLVs provides a rigorous understanding of the geometry and cohomology of Rapoport–Zink spaces for unitary groups at ramified primes. These geometric properties are fundamental for:

• Describing the supersingular (basic) locus in Shimura varieties and understanding their reduction modulo $p$.
• Calculating intersection numbers and verifying arithmetic fundamental lemmas within the framework of the Kudla–Rapoport program.
• Facilitating the paper of local and global Langlands correspondences via the geometry and $\ell$-adic cohomology of Rapoport–Zink spaces.

A plausible implication is that future advances in explicit intersection theory and the trace formulas for Shimura varieties with bad reduction will build upon the structural results about flatness, stratification, and Deligne–Lusztig correspondence established for Rapoport–Zink spaces of unitary type at deep level and ramified prime (Trentin, 2023).

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Summary Table: Structure of Irreducible Components in GU$(2,4)$ Rapoport–Zink Space

Lattice Type	Geometric Structure	Dimensionality
Maximal vertex lattice	Gen. Deligne–Lusztig variety for $\mathrm{Sp}_6$	5
$2$-modular lattice (split)	Line bundle over generalized DL variety ($\mathrm{SO}_6$)	4
$2$-modular lattice (non-split)	Closure of rank-2 bundle over classical DL variety ($\mathrm{SO}_6$)	4

This table organizes key correspondences between the lattice indexing of irreducible components, their geometric identification with (generalized) Deligne–Lusztig varieties, and their dimensions.

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In conclusion, the structure of Rapoport–Zink spaces for $\mathrm{GU}(2,4)$ at ramified primes is now precisely understood in terms of flatness, stratification by vertex and $2$-modular lattices, and geometric realization as (universal homeomorphisms to) Deligne–Lusztig varieties of symplectic and orthogonal type. Group-theoretic reduction techniques, rooted in the framework of admissible sets and affine Deligne–Lusztig theory, provide the underlying combinatorial and geometric machinery for this description—ultimately connecting $p$-adic geometry, arithmetic, and the representation theory of finite and $p$-adic groups in an explicit and calculable fashion.