Foliations in moduli spaces of abelian varieties

Oort, Frans

doi:10.1090/S0894-0347-04-00449-7

Foliations in moduli spaces of abelian varieties
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by Frans Oort;

J. Amer. Math. Soc. 17 (2004), 267-296

DOI: https://doi.org/10.1090/S0894-0347-04-00449-7

Published electronically: January 7, 2004

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Abstract:

We study moduli spaces of polarized abelian varieties in positive characteristic. Our final goal will be to understand Hecke orbits in such spaces. This paper provides one of the tools. For a given $p$-divisible group, all abelian varieties which give rise to this group have moduli points in a locally closed subset of the moduli space; we call an irreducible component of this subset a central leaf. Newton polygon strata are foliated by such leaves. Moreover, iterated $\alpha _p$-isogenies give a second leaf structure, which was already known under the name of Rapoport-Zink spaces. Any Newton polygon stratum is, up to a finite morphism, isomorphic to a product of an isogeny leaf and a finite cover of a central leaf. We conjecture that any Hecke-$\ell$-orbit is dense in the corresponding central leaf.

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