Ekedahl–Oort Strata

• Ekedahl–Oort strata are locally closed subsets of characteristic‑p moduli spaces defined by constant isomorphism classes of truncated p‑divisible groups.
• They are parameterized by finite Weyl-group data with dimensions measured by Coxeter length and exhibit smooth, quasi‑affine, and non‑empty geometry.
• These strata refine the Newton stratification in Shimura and Siegel varieties, linking theories of F‑zips, period maps, and intersection theory.

Ekedahl–Oort strata are the locally closed pieces of a characteristic-$p$ moduli space on which the isomorphism class of the truncated $p$-divisible group of level $1$ is constant. In the Siegel case they are the loci where $(A[p],\lambda[p])$ has fixed isomorphism class as a self-dual BT-$1$, while for Shimura varieties of Hodge type with good reduction at $p>2$ they are defined via the smooth zip period morphism to the stack of $G$-zips of type $\mu$. Their parameter set is finite and Weyl-group-theoretic, their dimensions are given by Coxeter length, and their closures are controlled by a twisted Bruhat-type order (Zhang, 2013).

1. Classical definition and conceptual role

Let $k$ be a field of characteristic $p>0$, and let $p$0 be a principally polarized abelian variety over $p$1 of dimension $p$2. The finite group scheme $p$3 is a finite, commutative group scheme of rank $p$4, and the principal polarization induces an isomorphism

$p$5

compatible with Cartier duality, so $p$6 becomes a self-dual BT-$p$7. Fixing the Siegel moduli scheme $p$8 over $p$9, the locus where $1$0 has a prescribed isomorphism class is an Ekedahl–Oort stratum. Oort showed that there are $1$1 such classes, that each corresponding stratum is nonempty and locally closed, that each stratum is quasi-affine, and that there is a dimension formula in terms of the combinatorics of the symplectic group (Zhang, 2013).

This stratification is finer than the Newton stratification. The Newton polygon records the isogeny class of the $1$2-divisible group $1$3, whereas the Ekedahl–Oort type records the isomorphism class of $1$4. Accordingly, each Newton stratum is a union of Ekedahl–Oort strata, and different Ekedahl–Oort types can occur inside a single Newton stratum. In the later group-theoretic formulation, Newton strata correspond to $1$5-conjugacy classes, while Ekedahl–Oort strata are indexed by Weyl-group data attached to parabolics and cocharacters (Zhang, 2013).

2. $1$6-zips, $1$7-zips, and Weyl-group parametrization

The modern formulation replaces the direct classification of BT-$1$8s by the theory of $1$9-zips and $(A[p],\lambda[p])$0-zips. An $(A[p],\lambda[p])$1-zip is a quadruple

$(A[p],\lambda[p])$2

consisting of a locally free module, descending and ascending filtrations, and Frobenius-linear isomorphisms between associated graded pieces. For abelian schemes in characteristic $(A[p],\lambda[p])$3, $(A[p],\lambda[p])$4 carries a canonical $(A[p],\lambda[p])$5-zip structure (Zhang, 2013).

Let $(A[p],\lambda[p])$6 be connected reductive and let $(A[p],\lambda[p])$7 be the cocharacter determined by the Shimura datum. Writing $(A[p],\lambda[p])$8 and $(A[p],\lambda[p])$9 for the parabolics defined by $1$0, with common Levi $1$1, a $1$2-zip of type $1$3 over a $1$4-scheme $1$5 is a quadruple

$1$6

where $1$7 is a right $1$8-torsor, $1$9 is a right $p>2$0-torsor, $p>2$1 is a right $p>2$2-torsor, and

$p>2$3

is an isomorphism of $p>2$4-torsors. These objects form a smooth Artin stack $p>2$5 of dimension $p>2$6, and Pink–Wedhorn–Ziegler identify it with a quotient stack

$p>2$7

for an explicit zip group $p>2$8 (Zhang, 2013).

Fixing a Borel $p>2$9, a maximal torus $G$0, Weyl group $G$1, and the subset $G$2 corresponding to the parabolic type of $G$3, one obtains the finite indexing set $G$4 of minimal-length representatives for $G$5. The $G$6-orbits on $G$7 are indexed by $G$8. For each $G$9, the corresponding orbit $\mu$0 is locally closed and smooth, its codimension is $\mu$1, and its closure is

$\mu$2

for a twisted Bruhat-type partial order $\mu$3 on $\mu$4 (Zhang, 2013).

3. Hodge-type Shimura varieties and the zip period morphism

Let $\mu$5 be a Shimura datum of Hodge type with hyperspecial level at a prime $\mu$6. By work of Vasiu and Kisin, the associated Shimura variety admits a smooth integral canonical model $\mu$7 over $\mu$8, characterized by the extension property. After choosing a symplectic embedding

$\mu$9

one obtains over the special fiber $k$0 a universal abelian scheme, its de Rham cohomology $k$1, the Hodge filtration, Frobenius and Verschiebung, and a system of tensors cutting out the $k$2-structure (Zhang, 2013).

From these data one constructs a $k$3-zip of type $k$4 on $k$5, hence a canonical morphism

$k$6

The Ekedahl–Oort stratum of type $k$7 is then

$k$8

Zhang proved that $k$9 is smooth. It follows that the strata are smooth and equidimensional of dimension $p>0$0 when nonempty, and that their closures are pulled back from the closure relations in $p>0$1 (Zhang, 2013).

The construction is intrinsic. The resulting morphism

$p>0$2

is independent of the choice of symplectic embedding, of the chosen integral model inside a symplectic representation, and of the chosen defining tensors; under a suitable integral extension hypothesis, it is also functorial for morphisms of Shimura data (Zhang, 2014). A complementary reinterpretation uses Breuil–Kisin windows: for the special fiber $p>0$3, one constructs a morphism

$p>0$4

to a quotient of a loop-group space, and its geometric fibers are exactly the Ekedahl–Oort strata, giving a loop-group realization of Viehmann’s truncations of level one (Yan, 2018).

4. Basic geometry: smoothness, purity, quasi-affineness, and non-emptiness

For Hodge-type Shimura varieties with good reduction at $p>0$5, the smoothness of $p>0$6 implies that the strata $p>0$7 are reduced, locally closed, smooth, and equidimensional, with

$p>0$8

when $p>0$9, and with closure formula

$p$00

In the Siegel case this recovers Oort’s stratification, and in PEL type it recovers the earlier Moonen–Wedhorn–Viehmann–Wedhorn picture (Zhang, 2013).

Purity was subsequently established in a general zip-theoretic form. The zip stratification attached to an arbitrary $p$01-zip is pure, and this implies purity of level-$p$02 stratifications for truncated Barsotti–Tate groups as well as purity of the Ekedahl–Oort stratification on special fibers of good models of Hodge-type Shimura varieties. The same work proves that all Ekedahl–Oort strata are quasi-affine schemes and extends several Bruhat-stratification properties from the PEL case to Hodge type (Wedhorn et al., 2014).

Further geometric refinements are available on compactifications. On the toroidal compactification, the strata defined by pullback of zip strata are expected to satisfy the same dimension and closure formulas; on the minimal compactification, the induced Ekedahl–Oort strata are affine, while the open-shimura-variety strata are quasi-affine (Wedhorn et al., 2018). In the PEL setting, Viehmann–Wedhorn proved that every Ekedahl–Oort stratum is smooth, quasi-affine, of dimension $p$03, and non-empty (Viehmann et al., 2010).

Non-emptiness in Hodge type admits a mod-$p$04 period-map proof. Using the perfectoid tower and the Hodge–Tate period map, Andreatta compared pullbacks of Ekedahl–Oort strata with pullbacks of fine Deligne–Lusztig strata on the special fiber of the flag variety, obtained an order-reversing Weyl-theoretic constraint on simultaneous intersections, and deduced that all Ekedahl–Oort strata are non-empty for $p$05 (Andreatta, 2021).

5. Relations to Newton strata, Bruhat geometry, period maps, and tautological rings

Ekedahl–Oort strata are part of a larger constellation of characteristic-$p$06 stratifications. Newton strata are determined by isocrystal data and are coarser; each Newton stratum is a union of Ekedahl–Oort strata. In the Hodge-type setting, the $p$07-ordinary locus coincides with the ordinary Ekedahl–Oort stratum and is open dense. Bruhat stratifications arise by forgetting part of the zip data, and in the Hodge-type setting the Bruhat stratification is defined through the same zip-period morphism $p$08 (Zhang, 2013).

The mod-$p$09 Hodge–Tate period map gives a different comparison. For a Hodge-type Shimura variety with perfectoid cover $p$10, there are two mod-$p$11 period maps, one to the special fiber of the Shimura variety and one to the special fiber of the flag variety. On the flag-variety side there is a fine Deligne–Lusztig stratification, and the two pullback stratifications are related by an order-reversing bijection between the relevant Weyl-theoretic indexing sets. This comparison gives a new proof of non-emptiness of all Ekedahl–Oort strata and places Ekedahl–Oort strata inside the same period-map framework that governs other $p$12-adic and mod-$p$13 stratifications (Andreatta, 2021).

Ekedahl–Oort strata also enter intersection theory. For the special fiber of a good reduction of a Hodge-type Shimura variety, the tautological ring is the subring of the Chow ring generated by all Chern classes of all automorphic bundles, and it is generated as a vector space by the cycle classes of the closures of the Ekedahl–Oort strata. Wedhorn and Ziegler compute these cycle classes, identify the Chow ring of the $p$14-zip stack with the rational cohomology ring of the compact dual, derive triviality of $p$15-adic Chern classes of flat automorphic bundles in characteristic $p$16, and obtain a new proof of Hirzebruch–Mumford proportionality for Shimura varieties of Hodge type (Wedhorn et al., 2018).

6. Examples, special cases, and singularity theory

In the Siegel case $p$17, the indexing set $p$18 has cardinality $p$19, and the general Hodge-type construction recovers Oort’s original Ekedahl–Oort stratification. In PEL type, the same Weyl-group indexing, the dimension formula $p$20, the closure order, and quasi-affineness were established in full generality, together with the notion of minimal Ekedahl–Oort strata and their relation to Newton strata. In the split PEL case, each Newton stratum contains a unique minimal Ekedahl–Oort stratum (Zhang, 2013, Viehmann et al., 2010).

A notable non-PEL example is given by CSpin Shimura varieties. For a quadratic space of signature $p$21, the associated Hodge-type Shimura datum yields an orthogonal Weyl-group combinatorics. If $p$22 is odd, the poset $p$23 is totally ordered and identified with $p$24 by the length function, so there is at most one stratum of each dimension $p$25. If $p$26 is even, there are at most $p$27 strata for $p$28, at most one stratum of each dimension $p$29, and at most two strata of dimension $p$30 (Zhang, 2013).

The geometry of closures has become a separate subject. Goldring and Koskivirta determined the normalization of closed Ekedahl–Oort strata via partial flag spaces: for each $p$31, a canonical parabolic $p$32 yields a minimal fine stratum in the associated partial flag space, and the normalization of the closure of the stratum of type $p$33 is obtained from the closure of that fine stratum (Koskivirta, 2017). More recently, singularity questions were treated for abelian-type Shimura varieties: there are conceptual and combinatorial criteria for normality and Cohen–Macaulayness of unions of strata, an exact criterion for when the union of two adjacent strata is smooth, explicit numerical criteria for one-dimensional stratum closures, and, for groups of type $p$34, a description of the smooth and normal loci of all closures together with reduced strata Hasse invariants and closed-form cycle-class formulas (Koskivirta et al., 19 Jun 2025).

Ekedahl–Oort stratifications also appear outside Shimura varieties through the Torelli map. The induced stratification on the moduli of curves of genus $p$35 has been studied in positive characteristic and in characteristic $p$36, with explicit constructions of families having prescribed Ekedahl–Oort type and dimension computations for the resulting loci (Zhou, 2018, Zhou, 2020).