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

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探索两种不同类型对象之间关系的富有成效的基础，一种来自分析世界，涉及无限和，另一种来自几何世界，涉及某些子空间如何相互相遇的知识。从Gross-Zagier的经典工作开始，这种关系的路径极大地增强了我们对椭圆曲线密码学的理解。本课题的目的是研究正交Shimura变上高协维环的交点理论，并着眼于Kudla关于交点数与某些Eisenstein级数的中心导数的猜想。这将涉及一个双管齐下的方法：首先，在积分模型上定义合适的生成周期系列，并显示它们的模块化。其次，建立Ekedahl-Oort地层的p进解析理论，该理论可与刚性解析交点理论的最新进展相结合，计算相应的局部交点。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.