Foliations in moduli spaces of abelian varieties

By Frans Oort. 2002

139 citations recorded by the source corpus.

In this paper we study abelian varieties and p-divisible groups in characteristic P Even though a non-trivial deformation of an abelian variety can produce a non trivial Galois-representation, say on the Tate-?-group of the generic fiber (in any characteristic), the geometric generic fiber has a "constant" Tate-^-group. We en counter the same phenomenon for the p-structure in positive characteristic: any two ordinary abelian varieties of the same dimension over an algebraically closed field have isomorphic p-divisible groups. However, for non-ordinary abelian varieties this seems to break down. The fas cinating structure which comes out of this is that the maximal locus where a given geometric isomorphism class of a p-divisible group is realized (e.g., in a family of abelian varieties) is a locally closed set. The locus defined by the geometric iso morphism type of a p-divisible group will be called a "central leaf"; see 2.1 and 3.4. This gives rise to a "foliation" of the open stratum attached to a Newton polygon ?; the dimension of any "leaf" in the same Newton polygon stratum solely depends on ?. In extreme cases either the leaf is the whole stratum, as in the ordinary case, or in the "almost ordinary case" (the p-rank equals g ? 1), or a leaf is zero-dimensional as in the supersingular case; in intermediate cases a leaf can be a proper subset and still be positive dimensional: we have worked out the example of g = 4 in 8.1.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d5b2ebdfb240ba22f60de86a6ac9aff0342976a

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Citations in the corpus

On the cohomology of certain PEL-type Shimura varieties (2005, background): “In [22], Oort introduces a foliation of the Newton polygon strata of moduli spaces of abelian varieties in positive characteristic p.”
On certain unitary group Shimura varieties (2004, background): “These are the leaves of Oort's foliation (see [26]).”
Crystalline Boundedness Principle (2002, background): “Only recently, this paper and [36] contain explicit analogues of (∗) for 2 d r − 2.”
The Newton stratification on deformations of local G-shtukas (2008, background): “There, work of Oort [O1] shows that up to a finite morphism the Newton stratum Nν is a product of the underlying scheme of a Rapoport-Zink space with a so-called central leaf, and by [O2], or [Ch2] the difference 〈2ρ, ν〉 between the dimensions is the dimension of this central leaf.”
Ekedahl–Oort and Newton strata for Shimura varieties of PEL type (2010, methodology): “language. We use Theorem 5 to show in the split case two results about the intersection of Newton and Ekedahl-Oort strata. In the Siegel case the ﬁrst one (Theorem 8.20) had been conjectured by Oort ([Oo3]) and proved by Harashita ([Har2]): Theorem 7. Assume that Gis split. Then for all w∈ JWwith Aw”
Dimensions of some affine Deligne–Lusztig varieties (2005, methodology): “1 in [26], which gives the dimensions of all central leaves in the Newton strata for the Siegel modular variety for GSp(8), with hyperspecial level structure at p.”
Stratifications in the reduction of Shimura varieties (2015, background): “(3) The fibers of ΥK are the group-theoretic version of Oort’s central leaves [38].”
Minimal p-divisible groups (2004, background): “Fol In [8] we define in the same moduli spaces a foliation : moduli points are in the same leaf if and only if the corresponding p-divisible groups are geometrically isomorphic; in this way we obtain a foliation of every open Newton polygon stratum.”
The \mu-ordinary locus for Shimura varieties of Hodge type (2013, methodology): “on SK(G,S±)⊗Fp is obtained by classifying the p-torsion subgroups Ax[p] up to isomorphism (as BT1-groups), see [Oo2]. Other examples of stratiﬁcations arise from considering Ax[p∞] up to isomorphism ([Oo3]), and from the p-rank of Ax[p] ([Kob]). The ordinary locus in SK(G,S±) ⊗Fp is deﬁned as the se”