Hilbert-Siegel moduli spaces in positive characteristic

Hilbert-Siegel varieties are moduli spaces for abelian varieties equipped with an action by an order OK in a fixed, totally real field K. As such, they include both the Siegel moduli spaces (use K = Q and the action is the standard one) and Hilbert-Blumenthal varieties (where the dimension of K is the same as that of the abelian varieties in question). In this paper we study certain phenomena associated to HilbertSiegel varieties in positive characteristic. Specifically, we show that ordinary points are dense in moduli spaces of mildly inseparably polarized abelian varieties with action by a given totally real field. Moreover, we introduce a combinatorial invariant of the first cohomology of an abelian variety which allows us to compute and explain the singularities of such a moduli space.