Moduli of abelian varieties

这个项目的重点“模的阿贝尔品种”是这些模品种的赫克对称性在一个领域的积极特征p.在领域的特征零赫克对称性支配模形式和他们的高维推广，以及p-adic性质通常反映在这些对称的几何性质中，在特征p中。固定模空间中给定点的Hecke对称产生局部稳定子群。这个群在局部模空间上的作用包含了关于一般赫克对称性的重要信息。然而，在正特征p的情况下如何提取这些信息还不清楚。最近，PI在上述一般问题的第一个非平凡情况下取得了一些进展，并发现了可以被认为是局部稳定子群在二维Lubin情形下作用的渐近展开式-Tate空间的特征p。PI提出将这种渐近展开扩展到其他模空间，并证明了在一般开牛顿层中的点的每一个Hecke轨道在与酉群相关的某些模簇中是稠密的。后一个问题被称为赫克轨道猜想，以前无法实现。这个建议还包含两个项目有关的Hecke对称性领域的特征零。 其中之一是继续PI先前对CM提升的支持研究;另一个是与Andrea-Oort猜想的另一个支持研究有关。对称性的概念起源于我们的基本审美意识，在现代科学中具有根本的重要性。在数学中，对称性的主要来源是由群的抽象定义所具体化的。主要研究对象在这个建议是某些家庭的系统多项式方程承认大量收集的对称性，所谓的赫克对称性，他们是根本的重要性，在数论。在该提案中，PI建立了一种方法，将揭示某些隐藏的模式，关于赫克对称性以前不知道。预计这些模式将使我们能够解决许多情况下的一个开放的问题称为赫克轨道猜想。 这种定性理解这些对称性的方法是自20世纪60年代中期Lubin和Tate的工作以来全新的。

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.