Foliations in moduli spaces of abelian varieties

In this paper we study abelian varieties and of p-divisible groups in characteristic p. Even though a non-trivial deformation of an abelian variety can produce a non-trivial Galois-representation, say on the Tate-l-group of the generic fiber (in any characteristic), the geometric generic fiber has a “constant” Tate-l-group. The same phenomenon we encounter for the p-structure in positive characteristic: any two ordinary abelian varieties of the same dimension over an algebraically closed field have isomorphic p-divisible groups. However, for non-ordinary abelian varieties this seems to break down. The fascinating structure which comes out of this is that the maximal locus where a given isomorphism class of a p-divisible group is realized (e.g. in a family of abelian varieties) is a locally closed set. We have called the locus defined by the geometric isomorphism type of a p-divisible group a “central leaf”, see (2.1), and (3.5). This gives rise to a “foliation” of the stratum attached to a Newton polygon ξ; the dimension of any “leaf” in the same Newton polygon stratum solely depends on ξ. In the extreme cases either the leaf is the whole stratum, as in the ordinary case, or a leaf is zero-dimensional as in the supersingular case; in intermediate cases a leaf can be a proper subset and still be positive dimensional: we have worked out the example of g = 4 in (8.1).