Flatness of the mod $p$ period morphism for the moduli space of   principally polarized abelian varieties
  In this note it is shown that associating to a principally polarized abelian variety its de Rham cohomology defines a faithfully flat morphism of the moduli space of principally polarized abelian varieties in positive characteristics to the moduli space of symplectic $F$-Zips as defined in \cite{MW}. 
Torsten Wedhorn