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Foliations in moduli spaces of abelian varieties


Author: Frans Oort
Translated by:
Journal: J. Amer. Math. Soc. 17 (2004), 267-296
MSC (2000): Primary 14K10, 14L05
Published electronically: January 7, 2004
MathSciNet review: 2051612
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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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Additional Information

Frans Oort
Affiliation: Mathematisch Instituut, Postbus 80.010, NL-3508 TA Utrecht, The Netherlands
Email: oort@math.uu.nl

DOI: http://dx.doi.org/10.1090/S0894-0347-04-00449-7
Received by editor(s): June 16, 2002
Published electronically: January 7, 2004
Article copyright: © Copyright 2004 American Mathematical Society