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Foliations in moduli spaces of abelian varieties. (English) Zbl 1041.14018
Fix a prime \(p\). While a non-trivial deformation of an abelian variety can produce a nontrivial Galois representation, the geometric generic fiber has a constant Tate \(\ell \)-group. In addition, two ordinary abelian varieties of the same dimension have isomorphic \(p\)-divisible groups over an algebraically closed field.
This paper is a study of what happens if the abelian varieties are non-ordinary. In this case the maximal locus where a given geometric isomorphism class of a \(p\)-divisible group is realized is a locally closed set. For a \(p\)-divisible group this locus is referred to as a “central leaf”. The term “foliation” refers to the “isogeny leaves” in the open stratum of a Newton polygon. Between two central leaves in the same stratum there is a correspondence by iterated \(\alpha _{p}\)-isogenies.
The author studies this foliation, paying specific attention to central leaves. The following theorems are proved (all from the paper):
1.
A finite level of a geometrically fiberwise constant family of \(p\) -divisible groups becomes constant over an appropriate finite cover of the base.
2.
A central leaf is closed in its open Newton polygon stratum.
3.
Central leaves are smooth over the base field, and the dimension of a central leaf depends only on the Newton polygon.
4.
For polarized abelian varieties, the union of all irreducible iterated \(\alpha _{p}\)-Hecke orbits through one point is a closed subset.
5.
Every component of a Newton polygon stratum is up to a finite morphism isomorphic with the product of any of the isogeny leaves with a finite cover of any of the central leaves.
While the first section introduces properties of \(p\)-divisible groups and finite group schemes (and proves the first result above), much of the background necessary for a reader who is not familiar with Newton polygons appears at the end of the paper. The next two sections are a treatment of central leaves, first the general case and then the polarized case. It is in these sections that the next two results are proved. Next is a study of isogeny leaves, which are irreducible components of certain subschemes of the module space of abelian schemes of fixed degree modulo the base field \(k\). It is here that the fourth result will be found. The last result appears in the next section, which deals with the product structure defined by the different types of leaves.
Finally, there are interesting questions and conjectures in the sixth section, which are to numerous to list here.

MSC:
14K10 Algebraic moduli of abelian varieties, classification
14L05 Formal groups, \(p\)-divisible groups
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References:
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