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-types on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of Goren and Oort [J. Alg. Geom., 2000] on the stratifications when p is unramified in O 𝐅 . We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when p is totally ramified in O 𝐅 .