---
title: Ekedahl-Oort Stratification
url: https://www.emergentmind.com/topics/ekedahl-oort-stratification
type: topic
---

# Ekedahl-Oort Stratification

Ekedahl-Oort stratification is the characteristic-\(p\) decomposition of a moduli space by the isomorphism class of the \(p\)-torsion group scheme, equivalently by the mod \(p\) reduction of the Dieudonné module or by de Rham cohomology endowed with Frobenius \(F\) and Verschiebung \(V\). In the classical setting of principally polarized abelian varieties it is encoded by final types or Young diagrams; on good reductions of Shimura varieties it is expressed through \(G\)-zips and Weyl-group combinatorics; via the Torelli morphism it induces stratifications on moduli of curves, and the formalism has been extended to stable curves, affine Deligne-Lusztig varieties, and several unitary and orthogonal Shimura-theoretic settings [1302.6261][1312.4869][2307.13445][2309.03371].

## 1. Classical definition and discrete invariants

For a principally polarized abelian variety \(A\) of dimension \(g\) over an algebraically closed field of characteristic \(p>0\), the Ekedahl-Oort type is a combinatorial invariant that classifies the principal quasi-polarized truncated Barsotti-Tate group scheme of level \(1\) formed by the \(p\)-torsion \(A[p]\) up to isomorphism. Equivalently, it classifies the Dieudonné module \(D(A[p])\) or the de Rham cohomology as a module under \(F\) and \(V\). In one standard presentation, the type is extracted from the final filtration on \(D(A[p])\): it is a list \(\nu=[\nu_1,\ldots,\nu_g]\), with \(\nu_i=\dim_k(V(N_i))\), subject to the combinatorial restrictions \(\nu_i\leq \nu_{i+1}\leq \nu_i+1\) [1302.6261].

A parallel description uses Young diagrams or sequences \(\mu=[\mu_1,\mu_2,\ldots,\mu_n]\), with \(g\geq \mu_1>\cdots>\mu_n>0\). In that notation, the \(p\)-rank is \(g-\mu_1\) and the \(a\)-number is \(n\). For a smooth curve \(C\), one defines \(\mu(C)\), \(\mathrm{p\text{-}rank}(C)\), and \(a(C)\) via the Jacobian \(J_C\). Moonen’s intrinsic formulation in terms of the Hasse-Witt triple
\[
(Q,\Phi,\Psi), \qquad Q=H^1(C,\mathcal O_C),\ \Phi=F,
\]
supplies a curve-theoretic realization of the same invariant and later becomes the basis for extending the notion to stable curves [2307.13445].

## 2. \(G\)-zips, Weyl groups, and Shimura varieties

For Shimura varieties of PEL type with good reduction, the Ekedahl-Oort strata are parameterized by the finite set
\[
{}^J W=\{w\in W\mid l(sw)>l(w),\ \forall s\in J\},
\]
where \(W\) is the Weyl group of the reductive group in the Shimura datum and \(J\subseteq I\) is determined by the datum. For every \(w\in {}^J W\), the corresponding stratum is non-empty, smooth, quasi-affine, and of dimension \(l(w)\), and the closure relation is
\[
\overline{\mathcal A_w}=\bigcup_{w'\preceq w}\mathcal A_{w'}.
\]
The same Weyl-group parameterization persists for good reductions of Shimura varieties of Hodge type, where one has a smooth morphism
\[
\zeta:\mathscr S_0\to G\text{-Zip}^{\mu}
\]
to the stack of \(G\)-zips of type \(\mu\), and the strata are the fibers of \(\zeta\) [1011.3230][1312.4869].

The zip-theoretic formulation makes the stratification intrinsic. The zip stratification given by an arbitrary \(\Ghat\)-zip over a scheme is pure; as a consequence, the Ekedahl-Oort stratification on special fibers of good models of Shimura varieties of Hodge type is pure, and all Ekedahl-Oort strata are quasi-affine schemes. In the Hodge-type setting the morphism \(\zeta\) is independent of the choice of symplectic embedding, and, under the assumption that a morphism of Shimura data extends to a morphism of reductive group schemes over \(\mathbb Z_p\), there is functoriality of Ekedahl-Oort stratifications with respect to morphisms of Shimura varieties [1404.0577][1401.6632].

A further refinement comes from period maps. For a Shimura variety of Hodge type admitting a smooth integral model at an odd prime \(p>3\), the perfectoid cover \(S^{\mathrm{ad}}(p^\infty)\) carries the Hodge-Tate period map to the flag variety. Pulling back the Ekedahl-Oort stratification along the specialization map \(q_1\) and the fine Deligne-Lusztig stratification along the specialization map \(q_2\), one obtains the comparison theorem: if
\[
q_1^{-1}(\mathrm{EO}(w'))\cap q_2^{-1}(\mathrm{DL}(w))\neq \emptyset,
\]
then \(w\leq_{\mathrm{DL}}\Omega(w')\), where \(\Omega(w)=w_0w\) is an order-reversing bijection. The same framework yields smoothness of \(\zeta\) and non-emptiness of all Ekedahl-Oort strata [2103.12361].

## 3. Purity, singularities, Hasse invariants, and cycle theory

Recent work has shifted attention from the existence and dimensions of strata to the singularities of their closures. For unions \(U\) of Ekedahl-Oort strata in the special fiber of abelian type Shimura varieties, there are conceptual and combinatorial criteria for normality and Cohen-Macaulayness in terms of canonical filtrations, canonical parabolics, and separating canonical covers. For an elementary \(w\)-open \(U=S_w\cup S_{w'}\), where \(S_{w'}\) has codimension one in the closure of \(S_w\), smoothness is equivalent to normality, to \(w\)-boundedness together with a separating canonical cover, and to extension of the canonical torsor. For one-dimensional Ekedahl-Oort strata closures, smoothness is determined by coincidence of canonical parabolics; in type \(A\), for \(G=\mathrm{GL}_n\) with signature \((r,s)\), the closure of the length-one stratum is smooth if and only if \(\gcd(r,s)=1\) [2506.16086].

For groups of type \(\mathsf B_n\), the structure is especially explicit. If
\[
S=\overline{S}_0\supseteq \overline{S}_1\supseteq \cdots \supseteq \overline{S}_{2n-1}
\]
is the Ekedahl-Oort filtration, then for \(0\leq j\leq n-1\) the smooth locus of \(\overline{S}_j\) is \(\bigcup_{i=j}^{2n-1-j}S_i\). Moreover, \(\overline{S}_j\) is normal, a local complete intersection, and admits a reduced Hasse invariant of weight \((p^{j+1}-1)\eta_\omega\), with cycle class
\[
[\overline{S}_j]=(p-1)(p^2-1)\cdots (p^j-1)[\omega].
\]
These results exhibit a precise link between the internal geometry of Ekedahl-Oort closures and automorphic line bundles [2506.16086].

The codimension-one singularity problem is now algorithmic. For an abelian type Shimura variety and an odd prime \(p\) of good reduction, regularity in codimension one of Zariski closures of Ekedahl-Oort strata is characterized in terms of the Frobenius action on the based root datum, and there is an algorithm that detects codimension-one singularities for arbitrary Ekedahl-Oort strata. In split type, the singularities are related to a stack of \(G\)-zips over \(\mathbb C\), and generalized Hasse invariants on this stack are governed by an explicit root-theoretic criterion [2603.17604].

The cycle-theoretic consequences are equally strong. Over \(\mathbb F_p\), the Chern classes of the Hodge bundle on \(A_g\) lift to the minimal compactification \(A_g^*\) in the best possible way: they are represented by algebraic cycles on \(A_g^*\otimes \mathbb F_p\) which define elements in its bivariant Chow ring. The mechanism is the Ekedahl-Oort stratification itself, whose closures provide effective representatives for tautological classes [1912.09687].

## 4. Curves, Jacobians, and induced Ekedahl-Oort strata

Via the Torelli morphism, the Ekedahl-Oort stratification on \(\mathcal A_g\) induces a stratification on \(\mathcal M_g\). This viewpoint extends beyond smooth curves. For a stable curve \(C\), the Hasse-Witt triple \((Q,\Phi,\Psi)\) with \(Q=H^1(C,\mathcal O_C)\) defines an intrinsic Ekedahl-Oort type, and this agrees with the Ekedahl-Oort type of the generalized Jacobian as a semi-abelian variety. The key comparison theorem states
\[
\mu(C)=\mu(J_C)=\mu(\widetilde C),
\]
where \(\widetilde C\) is the normalization. This extension allows dimension arguments for Ekedahl-Oort loci on compactified moduli of curves and generalizes known results for \(p\)-rank and \(a\)-number loci [2307.13445].

In characteristic \(2\), hyperelliptic curves admit a particularly rigid Ekedahl-Oort theory. If \(X\) is a hyperelliptic curve of genus \(g\) over an algebraically closed field of characteristic \(2\), then
\[
\HdR(X)\cong \HdR(E)^{\#B-1}\oplus \bigoplus_{\alpha\in B}\HdR(Y_\alpha),
\]
where \(E\) is an ordinary elliptic curve, \(B\) is the set of branch points, and \(Y_\alpha\) is an Artin-Schreier curve attached to \(\alpha\). Consequently,
\[
J_X[2]\simeq (\mathbb Z/2\oplus \mu_2)^r\oplus \bigoplus_{\alpha\in B}G_{c_\alpha},
\]
with \(r=\#B-1\) and \(c_\alpha=(d_\alpha-1)/2\). The isomorphism type depends only on the ramification invariants \(d_\alpha\), not on the locations of the branch points or the equation of \(X\), and the set of Ekedahl-Oort types that arise among hyperelliptic curves of genus \(g\) is in bijection with partitions of \(g+1\) [1007.1226].

For Hermitian curves \(X_q: y^q+y=x^{q+1}\), with \(q=p^n\), the Jacobian is supersingular and its Ekedahl-Oort type is completely determined. The distinct indecomposable factors of \(D(\mathrm{Jac}(X_q)[p])\) are in bijection with the orbits of the multiplication-by-two map on \(\mathbb Z/(2^n+1)-\{0\}\); an important feature is that these indecomposable factors do not depend on \(p\) [1302.6261].

Genus \(4\) supplies a laboratory for induced Ekedahl-Oort strata on moduli of curves. In characteristic \(3\), every smooth hyperelliptic curve of genus \(4\) has \(a\)-number at most \(2\), and for the hyperelliptic locus \(H_4\) the strata with \(u\leq [3,2,1]\) have the expected codimension, while several specified loci are irreducible. For general \(\mathcal M_4\) in characteristic \(3\), explicit families realize types such as \([3,2]\), \([3,2,1]\), and \([4,3]\), and for certain induced Ekedahl-Oort strata the codimension in \(\mathcal M_4\) equals the codimension of the corresponding stratum in \(\mathcal A_4\). At the same time, the superspecial stratum is empty because there are no superspecial curves of genus \(4\) in characteristic \(3\) [1812.04996][2003.12764].

## 5. Interaction with Newton strata and related stratifications

The Ekedahl-Oort and Newton stratifications are related but not equivalent. For Shimura varieties of PEL type, the Newton strata are parameterized by \(\sigma\)-conjugacy classes \(b\in B(G,\mu)\), and there is a group-theoretic notion of minimal Ekedahl-Oort stratum generalizing Oort’s definition in the Siegel case. In the split case, every Newton stratum contains a unique minimal Ekedahl-Oort stratum, and the intersection criterion is
\[
\mathcal A_w\cap N_b\neq \emptyset \iff w(b)\preceq w.
\]
The same analysis yields non-emptiness of all Newton strata, proving conjectures of Fargues and Rapoport generalizing Manin’s conjecture [1011.3230].

Unitary Shimura varieties of signature \((q-2,2)\) exhibit more intricate behavior. Their Ekedahl-Oort strata are indexed by permutations \(\gamma_{u,v}\) with \(1\leq u<v\leq q\), where \(\ell(\gamma_{u,v})=u+v-3\), and the closure order involves explicit relations beyond Bruhat order. Product maps, the forgetful map to the Siegel modular variety, and explicit Dieudonné theory provide criteria for which Ekedahl-Oort strata meet the supersingular locus. For the case \((3,2)\), this question is completely answered: some strata are contained in the supersingular locus, some meet it nontrivially, and others are disjoint from it [2405.04464].

For the \(\mathsf{GU}(3,2)\) Shimura variety, the picture is completely explicit. There are precisely \(10\) Ekedahl-Oort strata, labeled by
\[
\gamma_{1,2},\gamma_{1,3},\gamma_{1,4},\gamma_{1,5},\gamma_{2,3},\gamma_{2,4},\gamma_{2,5},\gamma_{3,4},\gamma_{3,5},\gamma_{4,5},
\]
and \(\dim \mathcal M(3,2)_{\gamma_{u,v}}=u+v-3\). There are four Newton strata, and \(\gamma_{3,4}\) is the unique Ekedahl-Oort stratum that intersects two Newton strata, namely \(\beta_1\) and the supersingular stratum; this is identified as the only nontrivial intersection of that kind [2510.01090].

In \(\mathcal A_5\), intersections of Ekedahl-Oort and Newton strata are also now largely explicit. Exactly four Ekedahl-Oort strata are fully contained in the supersingular locus \(\mathcal S_5\), and there are at most eight others with nontrivial intersection with \(\mathcal S_5\). For positive \(p\)-rank, all intersections in dimensions at most \(5\) are described by combining Oort’s minimality, Chai-Oort and Harashita’s results, and an inductive analysis of products of abelian varieties through the behavior of elementary sequences under direct sums [2509.19878].

A local analogue appears in affine Deligne-Lusztig varieties. For \(G=\mathrm{GL}_n\) and superbasic \(b\), the Ekedahl-Oort stratification
\[
X_\mu(b)=\bigsqcup_{w\in {}^S(\mu)}\pi(X_w(b))
\]
can be compared with the \(\mathbb J\)-stratification, or semi-module stratification. The latter refines the Ekedahl-Oort stratification if and only if a list of equivalent combinatorial conditions holds, including the existence, for every relevant \(w\), of \(v\in \mathrm{LP}(w)\) such that \(v^{-1}p(w)v\) is a Coxeter element, and this occurs exactly for an explicit finite list of cocharacters \(\mu\) up to central twist [2309.03371].

## 6. Embeddings, indecomposables, and bad-reduction phenomena

Ekedahl-Oort strata behave functorially under several natural embeddings of Shimura varieties. For embeddings between good reductions modulo \(p\) of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary Shimura varieties, the image of an Ekedahl-Oort stratum of the source is contained in a single Ekedahl-Oort stratum of the target. In the orthogonal case, the target index is determined by the source index, parity, and the position relative to the middle stratum; in the unitary case, the image rule depends on whether \(p\) is split or inert. The same work computes \(p\)-ranks and \(a\)-numbers: in the GSpin case this is done via the Kuga-Satake embedding, while in the unitary inert case the \(\mu\)-ordinary locus has \(p\)-rank \(2\), every other Ekedahl-Oort stratum has \(p\)-rank \(0\), the superspecial stratum has \(a\)-number \(n+1\), and every other stratum has \(a\)-number \(n-1\) [2605.27207].

At the level of individual unitary strata, there is now a complete classification of indecomposable Ekedahl-Oort strata for Shimura varieties associated to \(\mathsf{GU}(a,b)\) over an odd inert prime. Every indecomposable stratum is one of four types: unitary unicycle, unitary bicycle, Serre unicycle, or Serre bicycle. The classification is given in terms of primitive words in the alphabet \(\{\texttt f,\texttt v\}\); there is an algorithm translating such a description to the corresponding Weyl-group coset representative, and a tautological \(p\)-adic lift whose Newton polygon can be computed explicitly. As an application, the indecomposable strata corresponding to unitary unicycles and Serre unicycles always intersect the supersingular locus [2606.16882].

Bad reduction can produce sharply different behavior. For the Hodge-Tate period domain associated to a quaternionic Shimura curve at a ramified prime, the Newton stratification is trivial, but the Ekedahl-Oort stratification is not. There are three superspecial strata and two infinite families of non-superspecial strata, each indexed by \(k^\times\). Writing points as \(\tau\in X(C)\simeq C\cup\{\infty\}\), the superspecial strata are determined by intervals for \(\operatorname{ord}(\tau)\), while the non-superspecial strata at the critical values \(\operatorname{ord}(\tau)=\frac{p}{p+1}\) and \(\operatorname{ord}(\tau)=\frac{1}{p+1}\) are fibers of the maps
\[
\tau\mapsto \tau^{p+1}/p \bmod k,\qquad \tau\mapsto (p/\tau)^{p+1}\bmod k.
\]
This produces infinitely many Ekedahl-Oort strata, a pathology absent from the good-reduction case [1810.08814].

Source: https://www.emergentmind.com/topics/ekedahl-oort-stratification