 
Summary: HilbertSiegel moduli spaces in positive characteristic
Jeffrey D. Achter
achter@math.columbia.edu
April 19, 2001
HilbertSiegel 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 HilbertBlumenthal 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 Hilbert
Siegel 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.
The problem considered here arises in two distinct but closely related lines of inquiry. On one hand, recall
that if X is an abelian variety over a field k of characteristic p, then its ptorsion is described by X[p](k) =
(Z/pZ) for some . This integer , the prank, is between zero and dim X. When is maximal, the abelian
variety is said to be ordinary. Deuring shows that the generic elliptic curve is ordinary [4]. Mumford
announces [12], and Norman and Oort prove [14], the obvious generalization of this statement to higher
dimension: ordinary points are dense in the moduli space of polarized abelian varieties. Wedhorn has
recently obtained similar results [18] for families of principally polarized abelian varieties with given ring
of endomorphisms.
