Monodromy and irreducibility of leaves

Abstract

Conference on Abelian varieties, Amsterdam 29 - 31 May 2006. We show that non-supersingular Newton polygon strata in the principally polarized case are irreducible. Consider the theory of foliations of an open Newton polygon stratum in the moduli space of abelian varieties in positive characteristic. We show that any non-supersingular leaf is irreducible, and that the monodromy on such a leaf is maximal. Note that in the final result degrees of polarizations are arbitrary. The irreducibility of leaves, as proved here, is the discrete part of a proof of the Hecke orbit conjecture, which will be published in [7]. For a survey of this proof and for the terminology “discrete part” see [2].