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SUSTAINED p-DIVISIBLE GROUPS AND LEAVES ON MODULI SPACES OF ABELIAN VARIETIES

We explain a concept of of sustained p-divisible groups, discovered in collaboration with Frans Oort and motivated by the Hecke orbit problem. This concept leads to a scheme-theoretic definition of central leaves in moduli spaces of abelian varieties in characteristic p > 0, as well as central leaves in the space Def (Y0) of sustained deformations of a p-divisible group over a field of characteristic p. The formal completion C/z0 at any closed point z0 of a central leaf C in a moduli space of abelian varieties and the sustained deformation space Def (Y0) both have natural Tate-linear structure, that they are “built up” from a family of torsors for p-divisible formal groups. We also formulate a notion of strongly Tatelinear formal subschemes of Def (Y0), and a local rigidity question on whether every reduced and irreducible closed formal subscheme of Def (Y0) stable under a strongly non-trivial action of a p-adic Lie group is strongly Tate-linear.