Moduli of abelian varieties

阿贝尔簇的模

• 批准号: 1200271
• 负责人: Ching-Li Chai
• 金额: $ 32.51万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200271&HistoricalAwards=false
• 关键词: Moduli; abelian; varieties

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.

这个项目“阿贝尔变体的模”的重点是这些模变体在一个正特征p域上的Hecke对称性。在特征0域上，Hecke对称性支配着模形式及其高维推广。和p进性质通常反映在这些对称的几何性质中。在模空间中固定给定点的Hecke对称产生给定点的局部稳定子群。这个群在局部模空间上的作用包含了一般赫克对称的重要信息。然而，如何在正特征p的情况下提取这些信息是不清楚的。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，并发现了二维Lubin-Tate空间中局部稳定子群作用的渐近展开。PI提出将这种渐近展开推广到其他模空间。并证明了一般开牛顿地层中点的每一个赫克轨道在与酉群相关的某些模变体中是致密的。后一个问题被称为赫克轨道猜想，这在以前是无法实现的。该方案还包含两个与特征为零的域上的赫克对称相关的项目。其中一个继续PI先前支持的CM提升研究；另一个与另一项支持的关于Andrea-Oort猜想方面的研究有关。对称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，对称的主要来源是群的抽象定义。本建议的主要研究对象是某一类多项式方程组，它们包含大量的对称性，称为赫克对称；它们在数论中是至关重要的。在提案中，PI建立了一种方法，该方法将揭示以前不知道的赫克对称性的某些隐藏模式。预计这些模式将使我们能够解决被称为赫克轨道猜想的开放问题的许多情况。这种对这些对称性进行定性理解的方法是自20世纪60年代中期Lubin和Tate的工作以来全新的。