Ekedahl Oort Research Articles

Ekedahl–Oort strata of hyperelliptic curves in characteristic 2

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 group schemes which occur as the $2$-torsion group schemes of Jacobians of hyperelliptic $k$-curves of arbitrary genus.

Ekedahl–Oort and Newton strata for Shimura varieties of PEL type

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 using different methods. We determine the closure relations of the strata. We give a group-theoretical definition of minimal Ekedahl-Oort strata generalizing Oort's definition in the Siegel case and study the question whether each Newton stratum contains a minimal Ekedahl-Oort stratum. We give necessary criteria when a given Ekedahl-Oort stratum and a given Newton stratum meet. We determine which Newton strata are non-empty. This criterion proves conjectures by Fargues and by Rapoport generalizing a conjecture by Manin for the Siegel case.

Ekedahl–Oort strata and Kottwitz–Rapoport strata

We study Ekedahl–Oort strata on the moduli space A g of g-dimensional principally polarized abelian varieties in positive characteristic, and Kottwitz–Rapoport strata on its variants A J with parahoric level structure. First, we show that every Ekedahl–Oort stratum is isomorphic to a parahoric Kottwitz–Rapoport stratum. Second, both supersingular Ekedahl–Oort strata and supersingular Kottwitz–Rapoport strata are isomorphic to disjoint unions of Deligne–Lusztig varieties (see Hoeve (2010) [10] and Görtz and Yu (2010) [5], resp.), and here we compare these isomorphisms. Finally we give an explicit description of Kottwitz–Rapoport strata contained in the supersingular locus in the general parahoric case.

The supersingular locus in Siegel modular varieties with Iwahori level structure

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.

Ekedahl-Oort strata in the supersingular locus

The Ekedahl–Oort stratification is a stratification of the moduli space of abelian varieties in characteristic p>0. 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 apply this description to calculate the number of components of a supersingular Ekedahl–Oort stratum.

CONGRUENCE RELATIONS FOR SHIMURA VARIETIES ASSOCIATED TO SOME UNITARY GROUPS

In this paper we study the reduction to characteristic p of the Shimura variety associated to a unitary group which has signature (n−1, 1) at its real place. We describe the Newton polygon, the Ekedahl Oort, and the final stratification. In addition we examine the moduli space of p-isogenies using a variant of the local model for Shimura varieties. We apply our results to obtain a proof of the Eichler-Shimura congruence relation for the case that n is even.

The Ekedahl–Oort type of Jacobians of Hermitian curves

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 Dieudonne 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 = pn, which are supersingular curves well-known for their exceptional arithmetic properties. We determine the Ekedahl-Oort types of the Jacobians of all Hermitian curves. An interesting feature is that their indecomposable factors are determined by the orbits of the multiplication-by-two map on Z/(2n + 1), and thus do not depend on p. This yields applications about the decomposition of the Jacobians of Hermitian curves up to isomorphism.