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Hecke orbits on Shimura varieties of Hodge type

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Oort conjectured in 1995 that isogeny classes in the moduli space of principally polarised abelian varieties in characteristic p are Zariski dense in the Newton strata containing them. In this talk, we will present a proof of this conjecture under some minor hypotheses, (which in fact works for Shimura varieties of Hodge type). An important ingredient in our proof is a new theory of “Serre—Tate coordinates” on the formal deformation spaces of central leaves, in terms of so-called Dieudonn√©—Lie algebras. We also prove new results about monodromy groups of F-isocrystals for smooth varieties over a perfect field of characteristic p, which should be of independent interest. This is joint work with Marco D’Addezio.

This talk is part of the Number Theory Seminar series.

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