Let O F be the ring of integers of a totally real field F of degree g. We study the reduction of the moduli space of separably polarized abelian O F-varieties of dimension g modulo p for a fixed prime p. The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by a-numbers on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of Goren and Oort [GO] on… CONTINUE READING