Decidability of singularities in the Ekedahl--Oort stratification

1. Problem Statement

The paper addresses the geometric properties of the Ekedahl–Oort (EO) stratification on the special fiber (modulo $p$) of Shimura varieties of abelian type. Specifically, it focuses on the singularities that appear in the Zariski closures of these strata.

• Context: The EO stratification decomposes the special fiber $S$ of a Shimura variety into locally closed subvarieties indexed by a subset $I_W$ of the Weyl group $W$ of the underlying reductive group $G$.
• The Challenge: While the smoothness of individual strata is often understood, the singularities of their Zariski closures (unions of strata) are difficult to characterize. Determining whether a specific union of strata (specifically, "elementary $w$-open" subsets $U = X_w \cup X_{w'}$) is smooth or normal is a non-trivial problem with significant arithmetic implications (e.g., for constructing Galois representations).
• Goal: To provide an algorithm that decides the smoothness/normality of these closures and to relate these singularities in characteristic $p$ to objects defined in characteristic zero.

2. Methodology

The authors build upon the theory of $G$-zips (stacks parametrizing certain torsors with filtrations) and recent results from [KLPR25]. Their methodology involves three main pillars:

1. Reduction to Combinatorics of $G$-zips:

   • They utilize the smooth surjective map $\zeta: S \to X$ (the zip period map) from the Shimura variety to the stack of $G$-zips $X$.
   • The problem of singularities in $S$ is reduced to determining the smoothness of elementary $w$-open substacks $U(w, w') = X_w \cup X_{w'}$ in the stack $X$.
   • They rely on a criterion from [KLPR25] stating that $U(w, w')$ is smooth if and only if it is $w$-bounded (canonical parabolics satisfy $P_{w'} \subseteq P_w$) and admits a separating canonical cover.

2. Algorithmic Decidability via "Small" Elements:

   • The condition for a separating canonical cover depends on the image of a map $\pi_{\mathbb{F}_p}: W \to I_W$, which relates flag strata to zip strata.
   • Computing $\pi_{\mathbb{F}_p}$ generally is difficult. The authors introduce the notion of small elements ($W^{sm}$), defined as those $w \in W$ where the length is preserved: $\ell(\pi_{\mathbb{F}_p}(w)) = \ell(w)$.
   • They prove that for small elements, the map $\pi_{\mathbb{F}_p}$ coincides with an explicitly computable map $\Xi$ derived from the group structure.
   • They provide a combinatorial characterization of small elements in terms of roots (specifically, properties of "w-sequences" of roots), making the set $W^{sm}$ and the map $\pi_{\mathbb{F}_p}|_{W^{sm}}$ algorithmically computable.

3. Lifting to Characteristic Zero (Split Case):

   • For primes where $G$ splits, the authors construct a model of the zip stack over $\mathbb{Z}_p$ and its generic fiber over $\mathbb{Q}_p$ (and subsequently $\mathbb{C}$).
   • They define an analogous map $\pi_{\mathbb{C}}: W \to I_W$ for the complex stack.
   • Using Galois cohomology and reduction arguments, they show that the set of small elements and the map $\pi$ are independent of the exponent of the Frobenius (including the identity map $m=0$).
   • This allows them to transfer the decidability of singularities from characteristic $p$ to a characteristic zero object.

3. Key Contributions and Results

A. The Decidability Algorithm (Theorem A)

The paper provides a concrete algorithm to determine if an elementary $w$-open subset $U(w, w')$ is smooth (or normal).

• Input: A pair $(w, w')$ where $w'$ is a lower neighbor of $w$ in the partial order $\preceq_I$.
• Steps:
  1. Check $w$-boundedness: Verify if the type of the canonical parabolic of $w'$ is contained in that of $w$ ($I_{w'} \subseteq I_w$).
  2. Check the separating condition: Compute the set of small lower neighbors $\Gamma^{sm}_{I_w}(w)$. Verify that $w'$ is not in the image of the map $\pi_{\mathbb{F}_p}$ applied to $\Gamma^{sm}_{I_w}(w) \setminus \{w'\}$.
• Significance: This transforms a geometric question into a finite, combinatorial computation on the root datum of $G$.

B. Independence from the Prime (Theorem B)

The smoothness of these strata depends only on the action of the Frobenius on the based root datum $\Phi$.

• If two primes $p$ and $\ell$ induce the same automorphism $\sigma$ on the root datum (e.g., if $G$ splits at both), then the smoothness of the corresponding strata is identical.
• This implies that singularities are intrinsic to the Shimura datum's combinatorial structure, not the specific prime (provided the prime is of good reduction).

C. Application to Unitary Shimura Varieties (Theorem C)

The authors apply their algorithm to unitary Shimura varieties of signature $(r, s)$ at split primes.

• They completely classify the smoothness of 2-dimensional strata (length 2 elements in $I_W$).
• Result: Smoothness depends on $\gcd(r, s)$ and the modular inverse of $s$ modulo $n$.
  • If $\gcd(r, s) > 3$, the strata are singular.
  • If $\gcd(r, s) \in \{2, 3\}$, they are smooth.
  • If $\gcd(r, s) = 1$, smoothness depends on whether the modular inverse $m = \text{inv}_n(s)$ is greater or less than $n/2$.

D. Characteristic Zero Connection (Theorem D)

For split primes, the singularities in the EO stratification (characteristic $p$) are controlled by the geometry of the complex zip stack $X_{\mathbb{C}}$.

• $U(w, w')$ is smooth in characteristic $p$ if and only if the corresponding substack $U_{\mathbb{C}}(w, w')$ in the complex setting is $w$-bounded and admits a separating canonical cover.
• This suggests that the "bad" behavior in characteristic $p$ is encoded in the combinatorics of the complex group, opening the door to using complex analytic or differential geometric tools to study these singularities.

E. Generalized Hasse Invariants in Characteristic Zero (Theorem E)

The paper investigates the existence of generalized Hasse invariants (sections of line bundles vanishing exactly on the complement of a stratum) on the complex stack $X_{\mathbb{C}}$.

• Unlike in characteristic $p$, where Hasse invariants often require high weights, the authors show that in characteristic zero (specifically for the identity zip datum), trivial line bundles ($\lambda = 0$) can admit Hasse invariants.
• They provide a criterion for existence: A Hasse invariant exists for a stratum $w$ if there exists a weight $\lambda_0$ such that $\lambda = w\lambda_0 - z\lambda_0$ and $\langle \lambda_0, \alpha^\vee \rangle < 0$ for all "critical" roots $\alpha \in E_w$.
• Example: For $G=GL_n$ with signature $(n-1, 1)$, Hasse invariants of weight 0 exist on all strata.

4. Significance and Impact

1. Computational Arithmetic Geometry: The paper resolves a long-standing difficulty in determining the geometry of EO strata closures. By providing an algorithm, it enables the explicit computation of singularities for any Shimura variety of abelian type, facilitating the study of mod $p$ automorphic forms and Galois representations.
2. Bridging Characteristics: The discovery that singularities in characteristic $p$ are governed by a characteristic zero object ($X_{\mathbb{C}}$) is a profound conceptual shift. It suggests that the "pathology" of the EO stratification is not an artifact of the Frobenius map but a fundamental property of the underlying group structure.
3. New Tools for Moduli Spaces: The introduction of Hasse invariants in characteristic zero (and the "field with one element" heuristic) offers a new perspective on the geometry of moduli spaces, potentially linking the EO stratification to classical complex geometry and Hodge theory in novel ways.
4. Concrete Applications: The explicit classification for unitary Shimura varieties provides a testing ground for these theories and yields concrete results about the geometry of these specific moduli spaces, which are central to the Langlands program.

In summary, this paper establishes a rigorous, algorithmic framework for understanding the singularities of Ekedahl–Oort strata, proving their independence from the specific prime of reduction (under certain conditions), and linking them to a characteristic zero geometric object, thereby unifying arithmetic and complex geometric perspectives on Shimura varieties.