Flatness of the mod $p$ period morphism for the moduli space of principally polarized abelian varieties

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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}.
18 pages

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