P-adic Methods in the Arithmetic and Geometry of Shimura Varieties

志村品种算术和几何中的 P-adic 方法

• 批准号: 1802169
• 负责人: Keerthi Madapusi
• 金额: $ 14.17万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802169&HistoricalAwards=false
• 关键词: adic; Methods; Arithmetic; Geometry; Shimura

The main purpose of this project is to bring to bear new methods for the PI's ongoing study of certain spaces defined by polynomial equations, which have proven to be fruitful ground for exploiting relationships between two disparate kinds of objects, one from the world of analysis, involving infinite sums, and the other from the world of geometry, involving the knowledge of how certain subspaces meet each other. The path to such relationships, starting from the classical work of Gross-Zagier, has greatly enhanced our understanding of elliptic curve cryptography.The goal of this project is to study the intersection theory of higher codimension cycles on orthogonal Shimura varieties with an eye towards Kudla's conjectures relating intersection numbers with central derivatives of certain Eisenstein series. This will involve a two-pronged approach: First, to define suitable generating series of such cycles on integral models, and to show their modularity. Second, to develop a p-adic analytic theory of Ekedahl-Oort strata, which can be combined with recent advances in a rigid analytic intersection theory to compute the relevant local intersections.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的主要目的是为PI正在进行的由多项式方程定义的某些空间的研究带来新的方法，这些方法已被证明是利用两种不同对象之间关系的富有成效的基础，一种来自分析世界，涉及无限总和，另一种来自几何世界，涉及某些子空间如何相互满足的知识。从Gross-Zagier的经典工作开始，这种关系的路径大大提高了我们对椭圆曲线密码学的理解。本项目的目标是研究正交Shimura簇上的高余维循环的相交理论，着眼于Kudla的与某些Eisenstein级数的中心导数相关的相交数的结构。这将涉及到一个双管齐下的方法：首先，定义合适的生成系列的积分模型，并显示其模块化。第二，发展Ekedahl-Oort地层的p-adic分析理论，该理论可以与刚性分析交叉理论的最新进展相结合，以计算相关的局部交叉点。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。