Moduli of abelian varieties

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The focus of this project "Moduli of Abelian Varieties" is on the Hecke symmetry of these moduli varieties over a field of positive characteristic p. Over fields of characteristic zero the Hecke symmetries govern modular forms and their higher dimensional generalizations, and p-adic properties are often reflected in the geometric properties of these symmetries in characteristic p. Hecke symmetries which fix a given point in a moduli space give rise to the local stabilizer subgroup of the given point. The action of this group on the local moduli space contains crucial information about the Hecke symmetries in general. However it was unclear how to extract these information in the case of positive characteristic p. Recently the PI made some progress in the first non-trivial case of the above general problem, and found what can be thought of as an asymptotic expansion of the action of the local stabilizer subgroup in the case of two-dimensional Lubin-Tate space in characteristic p. The PI proposes to extend such asymptotic expansion to other moduli spaces, and to show that every Hecke orbit of a point in the generic open Newton stratum is dense in certain modular varieties associated to unitary groups. The latter problem is known as the Hecke orbit conjecture, which was inaccessible before. This proposal also contains two projects related to Hecke symmetries over fields of characteristic zero. One of them continues the PI's prior supported research on CM lifting; the other is related to another supported research on aspects of the Andrea-Oort conjecture.The concept of symmetry originated from our basic aesthetic sense and is of fundamental importance in modern science. In mathematics the major source of symmetry is crystallized by the abstract definition of a group. The main object of study in this proposal is a certain families of systems of polynomial equations admitting a large collection of symmetries, called Hecke symmetries; they are of fundamental importance in number theory. In the proposal, the PI establishes an approach which will reveal certain hidden pattern about the Hecke symmetries not previously known. It is expected that these patterns will enable us to solve many cases of an open problem known as the Hecke orbit conjecture. This approach to a qualitative understanding of these symmetries is completely new since the work of Lubin and Tate in the middle 1960's.