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

这个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，这已被证明是利用两种不同类型的对象之间的关系的丰硕基础，一种来自分析世界，涉及无限和，另一种来自几何世界，涉及某些子空间如何彼此相交的知识。本文从Gross-Zagier的经典工作出发，研究了正交Shimura簇上高余维圈的交理论，重点研究了Kudla猜想中交数与某些Eisenstein级数的中心导数之间的关系。这将涉及一个双管齐下的方法：首先，在整体模型上定义合适的此类周期的生成序列，并展示它们的模块化。第二，发展Ekedahl-Oort地层的p元分析理论，该理论可以与刚性解析交集理论的最新进展相结合来计算相关的局部交点。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.