Hecke orbits in moduli spaces
-- Frans Oort

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The Zariski closure of the Hecke orbit of 
a point in the moduli space of abelian
varieties in positive characteristic equals the related
Newton polygon stratum. This was proved by Ching-Li Chai
in 1995 for ordinary abelian varieties. The general case
was conjectured in 1993 by FO, and recently proved by
C.-L. Chai and FO. I will present a survey and one of the 
details of the proof.