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Hilbert-Siegel moduli spaces in positive characteristic Jeffrey D. Achter

Summary: Hilbert-Siegel moduli spaces in positive characteristic
Jeffrey D. Achter
April 19, 2001
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 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 p-torsion is described by X[p](k) =
(Z/pZ) for some . This integer , the p-rank, 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.


Source: Achter, Jeff - Department of Mathematics, Colorado State University
Chai, Ching-Li - Department of Mathematics, University of Pennsylvania


Collections: Environmental Sciences and Ecology; Mathematics