# Leaves in moduli spaces in characteristic p

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Frans Oort, University of Utrecht and Columbia University
Fine Hall 322

We try to understand the geometry of the moduli space of polarized abelian varieties in characteristic p. E.g. the phenomenon that Hecke orbits blow up and down in a rather unpredictable way. Choose a point $x$, corresponding to a polarized abelian variety. We study $C(x)$ consisting of all moduli points of polarized abelian varieties which have the same $p$-adic and $\ell$-adic invariants. This turns out to be a locally closed subset. We discuss properties of these sets, which form a foliation of the related Newton polygon stratum. We give several applications.