Ben Moonen
We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by cosets in a Weyl group. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.
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We study the induced Ekedahl-Oort stratification on the moduli space $\mathcal{M}_4$ of curves of genus $4$ in characteristic $3$. By constructing families of curves with given Ekedahl-Oort type and by computing the dimension of the intersection of the induced Ekedahl-Oort strata and the boundary divisor classes, we show that for certain induced Ekedahl-Oort strata in $\mathcal{M}_4$, they have the same codimension in $\mathcal{M}_4$ as the corresponding Ekedahl-Oort strata in moduli space $\mathcal{A}_4$ of principally polarized abelian varieties of dimension $4$.
We study the induced Ekedahl-Oort stratification on the moduli of curves of genus 4 in positive characteristic.
This short paper is a continuation of the author's Ph.D thesis, where Ekedahl-Oort strata are defined and studied for Shimura varieties of Hodge type. The main results here are as follows. 1. The Ekedahl-Oort stratification is independent of the choices of symplectic embeddings. 2. Under certain reasonable assumptions, there is certain functoriality for Ekedahl-Oort stratifications with respect to morphisms of Shimura varieties.
We give a description of the individual Ekedahl-Oort strata contained in the supersingular locus in terms of Deligne-Lusztig varieties, refining a result of Harashita.
We construct canonical non-vanishing global sections of powers of the Hodge bundle on each Ekedahl-Oort stratum of a Hodge type Shimura variety. In particular we recover the quasi-affineness of the Ekedahl-Oort strata. In the projective case, this gives a very short proof of non-emptiness of Ekedahl-Oort strata. It follows that the Newton strata are also nonempty, by a result of S.Nie. We also deduce the fact that the $μ$-ordinary locus is determined by the Ekedahl-Oort strata of its image by any embedding.
We study the Ekedahl-Oort stratification for good reductions of Shimura varieties of PEL type. These generalize the Ekedahl-Oort strata defined and studied by Oort for the moduli space of principally polarized abelian varieties (the "Siegel case"). They are parameterized by certain elements w in the Weyl group of the reductive group of the Shimura datum. We show that for every such w the corresponding Ekedahl-Oort stratum is smooth, quasi-affine, and of dimension l(w) (and in particular non-empty). Some of these results have previously been obtained by Moonen, Vasiu, and the second author usin…
We study the moduli space A_g of g-dimensional principally polarized abelian varieties in positive characteristic, and its variant A_I with Iwahori level structure. Both supersingular Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are isomorphic to disjoint unions of Deligne-Lusztig varieties (see [Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these isomorphisms. We also give an explicit description of Kottwitz-Rapoport strata contained in the supersingular locus in the general parahoric case. Finally, we show that every Ekedahl-Oort stratum is isomorphic to a para…
For a Shimura variety of Hodge type with hyperspecial level structure at a prime $p$, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p>2$. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn and Viehmann on good reductions of Shimura varieties of PEL type. …
Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime p>3. Consider its perfectoid cover S(p^\infty) and the Hodge-Tate period map introduced by A. Caraiani and P. Scholze. We compare the pull-back to S(p^\infty) of the Ekedahl-Oort stratification on the mod p special fiber of S and the pull back to S(p^\infty) of the fine Deligne-Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge-Tate period map. An application to the non-emptiness of Ekedhal-Oort strata is provided.
We develop a method to compute the Ekedahl-Oort type of a curve C over a field k of characteristic p (which is the isomorphism type of the p-kernel group scheme J[p], where J is the Jacobian of C). Part of our method is general, in that we introduce the new notion of a Hasse-Witt triple, which re-encodes in a useful way the information contained in the Dieudonne module of J[p]. For complete intersection curves we then give a simple method to compute this Hasse-Witt triple. An implementation of this method is available in Magma.
Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p=2$. We prove that the de Rham cohomology of $X$ decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the $2$-torsion group scheme $J_X[2]$ of the Jacobian of $X$ in terms of the Ekedahl-Oort type. The interesting feature is that $J_X[2]$ depends only on some discrete invariants of $X$, namely, on the ramification invariants associated with the branch points. We give a complete classification of the gr…
Let $ S $ be the special fibre of the good reduction of a Shimura variety of Hodge type. By constructing adapted deformations for the associated $p$-divisible groups of $ S $, we manage to construct a morphism from $S$ to some quotient sheaf of the loop group associated with $S$. We show that the geometric fibres of this morphism give back the Ekedahl-Oort strata of $S$. For any geometric point $ x $ of $ S $, we give a deformation over $ W(k(x)) $ of the $ p $-divisible group associated with $ x $ by (non-canonically) constructing a Breuil-Kisin window (which corresponds to a $ p $-divisible …
Ekedahl-Oort types of stable curves
本文将Ekedahl-Oort类型定义扩展至稳定曲线，证明其与广义雅可比的定义一致，并推广了关于曲线p-阶和a数位 locus 维数的结果。
Decidability of singularities in the Ekedahl--Oort stratification
研究通过Frobenius作用于根datum来刻画Abel型Shimura变元中Ekedahl-Oort分层的codimension一正则性，并提出算法检测任意分层的奇点，同时探讨split型Shimura数据下分层奇点与G-zip堆栈的关系及广义Hasse不变量的存在性。
Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$-covers in characteristic $2$
本文研究特征2中Z/2Z覆盖的Ekedahl-Oort类型，通过发展第二种增强微分理论，揭示了覆盖类型与底曲线性质之间的关系。
The number of Ekedahl-Oort strata intersecting the supersingular locus in the Siegel modular variety
本文证明了 Siegel 模簇中与超奇异轨迹相交的 Ekedahl-Oort 层数 $N_g$ 在 $g$ 趋于无穷时渐近等于 $2^{g-1}$。
Supersingular Ekedahl-Oort strata and Oort's conjecture
本文证明了在特定条件下，奇点Ekedahl-Oort层的几何通用成员的自同构群为$\{ \pm 1\}$，并验证了Oort猜想在偶数维和素数$p\geq 5$时的正确性。
Ekedahl-Oort Types and Newton Polygons of Abelian Covers of $\mathbf{P}^1$ Branched at Three Points
本文研究了$\mathbf{P}^1$在三个点处分支的阿贝尔覆盖的牛顿多边形与埃克萨尔-奥尔特类型，揭示了超奇异和超特殊曲线的高频出现，并为Oort猜想提供了新的证据。
The Ekedahl-Oort Stratification and the Semi-Module Stratification
本文在超基本情形下比较了仿射Deligne-Lusztig簇的$\mathbb J$-分层与Ekedahl-Oort分层，并分类了前者对后者进行细化且簇具有简单几何结构的情形。
We apply our theory of partial flag spaces developed in previous articles to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti-Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl-Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.
The Ekedahl-Oort type is a combinatorial invariant of a principally polarized abelian variety $A$ defined over an algebraically closed field of characteristic $p > 0$. It characterizes the $p$-torsion group scheme of $A$ up to isomorphism. Equivalently, it characterizes (the mod $p$ reduction of) the Dieudonné module of $A$ or the de Rham cohomology of $A$ as modules under the Frobenius and Vershiebung operators. There are very few results about which Ekedahl-Oort types occur for Jacobians of curves. In this paper, we consider the class of Hermitian curves, indexed by a prime power $q=p^n$, wh…
Intersections of the automorphism and the Ekedahl-Oort strata in $M_2$
研究二型曲线模空间中自同构strata与Torelli映射下埃克尔-奥尔特strata的交集，分析自同构群的可能结构及参数化方法，计算Jacobian变种的固定埃克尔-奥尔特类型strata的维度和不可约分量数量。
We show that the zip stratification given by an arbitrary $\Ghat$-zip over a scheme is pure. We deduce purity of the level-$m$-stratification for truncated Barsotti-Tate groups and purity of the Ekedahl-Oort stratification for special fibers of good models of Shimura varieties of Hodge type. We also show that all Ekedahl-Oort strata are quasi-affine schemes and generalize several properties of the Bruhat stratification from the PEL case to the case of Shimura varieties of Hodge type.
The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge bundle over $A_g$ has its Chern classes in the Chow ring of $A_g$ with rational coefficients. We show that over the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so in the best possible way: despite the highly singular nature of $A_g^*$ they are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements in its bivariant Chow ring. This is in contrast to the si…
Intersections of the Ekedahl-Oort and Newton Strata of $\mathcal{A}_{5}$
本文研究了$\mathcal{A}_{5}$的Ekedahl-Oort与牛顿层的交集，确定了五维情况下非空交集的结构，并给出了超奇异层上的诱导Ekedahl-Oort分层描述。
Ekedahl-Oort strata in the $\mathsf{GU}(q-2,2)$ Shimura variety
本文针对$\textsf{GU}(q-2,2)$志村簇的特征p纤维，提出若干新颖技术，以深化对一般签名下Ekedahl-Oort分层及其与Newton分层相互作用的理解。
The Ekedahl-Oort and Newton stratification of the $\mathsf{GU}(3,2)$ Shimura variety
本文研究$\textsf{GU}(3,2)$志村簇的特征$p$纤维，完整刻画了Ekedahl-Oort分层与Newton分层的相互作用，明确了两类分层中相交的层。
Ekedahl-Oort strata under natural embeddings of orthogonal and unitary Shimura varieties
研究 GSpin Shimura 簇与 Rapoport-Smithling-Zhang 酉 Shimura 簇在好约化之间的自然嵌入下，Ekedahl-Oort 分层的像，并计算这些 Shimura 簇的 p-秩和 a-数等离散不变量。
Ekedahl-Oort strata meeting the supersingular locus
本文给出了特征p>2时主极化阿贝尔簇模栈中EO层与超奇异轨迹相交的显式组合判据，并证明了渐近几乎所有p-秩为零的EO层均相交，同时应用于酉PEL Shimura簇给出下界。
Singularities in the Ekedahl--Oort stratification
本文研究阿贝尔型 Shimura 簇特殊纤维中 Ekedahl--Oort 层并集的正规性、Cohen--Macaulay 性及光滑性，给出判据、数值条件、B_n 型群的光滑与正规轨迹描述，并构造约化层 Hasse 不变量证明闭包为局部完全交集且提供周类公式。
Abelian covers of $\mathbb{P}^1$ of $p$-ordinary Ekedahl-Oort type
本文通过 Deligne--Mostow Shimura 簇研究 $\mathbb{P}^1$ 阿贝尔覆盖在素数 $p$ 处的 Ekedahl-Oort 型与 Newton 多边形下界，证明分支点至多五个且 $p$ 充分大时下界最优，推广了 Bouw 的 $p$-秩结果。
Tautological rings of Shimura varieties and cycle classes of Ekedahl-Oort strata
该研究定义了志村簇的tautological环，阐明Hodge型志村簇良好约化特殊纤维上该环的结构并证明其由Ekedahl-Oort层闭链类生成，还计算了这些闭链类并给出三项相关应用。
Given a principally polarized abelian variety $A$ of dimension $g$ over an algebraically closed field $k$ of characteristic $p$, the $p$ torsion $A[p]$ is a finite flat $p$-torsion group scheme of rank $p^{2g}$. There are exactly $2^g$ possible group schemes that can occur as some such $A[p]$. In this paper, we study which group schemes can occur as $J[p]$, where $J$ is the Jacobian of a hyperelliptic curve defined over $\mathbb{F}_p$. We do this by computing explicit formulae for the action of Frobenius and its dual on the de Rham cohomology of a hyperelliptic curve with respect to a given ba…
We generalize the notion of Ekedahl-Oort strata to elements in the loop group of any connected reductive group, and call the resulting discrete invariant the truncation of level 1 of the element. We give conditions for the Newton points occurring among the elements of a given truncation of level 1 and especially for the generic Newton point in a given truncation stratum. We prove that truncation strata are locally closed and give a description of the closure of each stratum. We also translate our results back to the original Ekedahl-Oort stratification of the reduction modulo p of Shimura vari…
We complete the study of the supersingular locus in the fiber at $p$ of a Shimura variety attached to a unitary similitude group $\GU(1,n-1)$ over $\QQ$ in the case that $p$ is inert. This was started by the first author in \cite{Vo_Uni} where complete results were obtained for $n =2,3$. The supersingular locus is uniformized by a formal scheme $\Ncal$ which is a moduli space of so-called unitary $p$-divisible groups. It depends on the choice of a unitary isocrystal $\Nbf$. We define a stratification of $\Ncal$ indexed by vertices of the Bruhat-Tits building attached to the reductive group of …
In the paper four stratifications in the reduction modulo $p$ of a general Shimura variety are studied: the Newton stratification, the Kottwitz-Rapoport stratification, the Ekedahl-Oort stratification and the Ekedahl-Kottwitz-Oort-Rapoport stratification. We formulate a system of axioms and show that these imply non-emptiness statements and closure relation statements concerning these various stratifications. These axioms are satisfied in the Siegel case.
In this paper a number of results on cycles on the moduli space of principally polarized abelian varieties is presented. Results include a determination of the tautological ring, bounds on the order of torsion of the top Chern class $λ_g$ and a determination of the cycle classes of the Ekedahl-Oort stratification in characteristic $p$. It includes as special cases formulas for the classes of loci like the $p$-rank $\leq f$ locus or the $a$-number $\geq a$-locus. The results on the tautological ring are my own work, the results on the torsion of $λ_g$ and on the cycle classes of the Ekedahl-Oor…
We describe the intersection of the Torelli locus $j(\mathcal{M}_4^{ct}) = \mathcal{J}_4$ with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic two. We show that the locus of supersingular Jacobians $\mathcal{S}_4\cap\mathcal{J}_4$ in characteristic two is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over $\mathbb{F}_2$, and another involves studying the Ekedahl-Oort types and using the indecomposability of some $\mathrm{BT}_1$ group schemes.
We review the results on the cycle classes of the strata defined by the height and the Artin invariant on the moduli of K3 surfaces in positive characteristic obtained in joint work with Katsura and Ekedahl. In addition we prove a new irreducibility result for these strata.
We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^μ$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags $G$-ZipFlag$^μ$, fibered in flag varieties over $G$-Zip$^μ$. It provides a simultaneous generalization of the "classical case" homogeneous complex manifolds studied by Griffiths-Schmid and the "flag space" for Siegel varieties studied by Ekedahl-van der Geer. Four …
This paper provides explicit closed formulas in terms of tautological classes for the cycle classes of the height and Artin invariant strata in families of K3 surfaces. The proof is uniform for all strata and uses a flag space as the computations in [arXiv:math/0412272] for the Ekedahl-Oort strata for families of abelian varieties, but employs a Pieri formula formula to determine the push down to the base space.
Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. We compare three classifications of $BT_1$ group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl--Oort types of Fermat quotient curves and we compute four invariants of the $p$-torsion group schemes of these curves.
We define quotient stacks of "zip flags". They form towers above the stack of $G$-zips introduced by Moonen, Pink, Wedhorn and Ziegler. We define a stratification on the stack of zip flags, and prove that it is principally pure, under a certain assumption on $p$. The fiber product with a Shimura variety of Hodge-type is a generalization of flag spaces considered by Ekedahl-Van der Geer. For large $p$, we prove that all strata are affine. We prove a theorem on discreteness of fibers for finite morphisms between stacks of $G$-zips. This allows us to prove that the zip stratification is principal…
We study the moduli space of abelian threefolds with Iwahori level structure in positive characteristic. We explicitly determine the fibers of the canonical projection to the moduli space of principally polarized abelian varieties and draw conclusions about the relationship between the Ekedahl-Oort, the Kottwitz-Rapoport and the Newton stratification on these spaces.
We review the Newton stratification and Ekedahl-Oort stratification on the special fiber of a smooth integral model for a Shimura variety of Hodge type at a prime of good reduction. We show that the μ-ordinary locus coincides with the generic Ekedahl-Oort stratum, and that for any two geometric points in the μ-ordinary locus there is an isomorphism of the attached Dieudonne modules with additional structure. As a consequence, we proof that the μ-ordinary locus is open and dense, thus generalizing the results which were already known in the PEL-case. To prove our results we provide a method whi…
We investigate the geometry of the special fiber of the integral model of a Shimura variety with parahoric level at a given prime place. To be more precise, we deal with the EKOR stratification which interpolates between the Ekedahl-Oort and Kottwitz-Rapoport stratifications. In the Siegel case we give a geometric description by suitably generalizing the theory of $G$-zips of Moonen, Wedhorn, Pink and Ziegler to our context.
In this paper, we construct the almost product structure of the minimal Newton stratum in deformation spaces of Barsotti-Tate groups with crystalline Tate tensors, similar to Oort's and Mantovan's construction for Shimura varieties of PEL-type. It allows us to describe the geometry of the Newton stratum in terms of the geometry of two simpler objects, the central leaf and the isogeny leaf. This yields the dimension and the closure relations of the Newton strata in the deformation space. In particular, their nonemptiness shows that a generalisation of Grothendieck's conjecture of deformations o…
The main goal of this paper is to generalize Serre-Tate theory of "ordinary" local moduli to Shimura varieties of PEL type. To this end we develop a generalized notion of ordinariness, we prove a number of basic results about this, and we study the formal deformations of ordinary objects. In general, the formal deformation spaces get a new "group-like" structure that we call a "cascade". Further the paper contains some results on the Ekedahl-Oort stratification of PEL moduli spaces, and an application of our results to the study of congruence relations.
In this paper we study the reduction of PEL-Shimura varieties associated to unitary groups of signature (n-1,1) in the inert and unramified case. We describe the Newton polygon and the Ekedahl-Oort stratification. We further study the associated moduli space of isogenies and use a variant of the local model to give a generic description. As an application we prove a conjecture of Blasius und Rogawski about the congruence relation if the integer n is even.
Brauer groups of abelian varieties over fields of finite characteristic
本文研究了有限特征域上阿贝尔簇的布雷厄姆群，重点分析了 p-原torsion 部分的结构，并通过同源类分类和埃克达尔-奥尔特类型计算了维数。
Level $m$-stratifications on PEL Shimura varieties are defined and studied by Wedhorn using BT-$m$s with PEL structure, and then by Vasiu for general Hodge type Shimura varieties using Shimura $F$-crystals. The theory of foliations is established by Oort for Siegel modular varieties, and by Mantovan for PEL Shimura varieties. It plays an important role in Hamacher's work to compute the dimension of Newton strata of PEL Shimura varieties. We study level $m$ stratifications on good reductions at $p>2$ of Shimura varieties of Hodge type by constructing certain torsors together with equivariant mo…
In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…
We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case $g$ is even.
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