Higher Codimension Cycles on Shimura Varieties

这个项目的主要重点是志村品种的理论。这些是特殊的高维曲面，其丰富的几何和算术使它们成为现代数学研究的中心对象之一。除了在纯数学中的重要性外，它们在朗兰兹程序中也发挥着重要作用，朗兰兹程序在理论物理学中日益引起人们的兴趣，它们是理解椭圆曲线和阿贝尔变分的重要工具，而椭圆曲线和阿贝尔变分现在在密码学中发挥着重要作用。更广泛地说，该项目涉及数论和算术几何。这是一个研究领域，通过密码学和编码理论的某些方面应用于网络安全。获得该奖项支持的研究生将接受培训，为这些项目做出贡献。首席研究员项目的主要目标是证明特殊循环和自同构形式之间的新关系。例如，首席研究员将利用Borcherds积理论和积分模型的最新进展来证明正交Shimura变量的积分模型上由任意余维的特殊循环形成的生成级数的模块化。首席研究员还将证明广义Gross-Zagier公式的新例子，通过用自同构l函数的积分核系数来表示酉Shimura变上循环的交多重。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 focus of this project is on the theory of Shimura varieties. These are particular kinds of higher-dimensional surfaces whose rich geometry and arithmetic puts them among the central objects of study in modern mathematics. In addition to their importance in pure mathematics, they play an essential role in the Langlands program, which is of increasing interest in theoretical physics, and are important tools for understanding elliptic curves and abelian varieties, which now play a major role in cryptography. More broadly, the project concerns the theory of numbers and arithmetic geometry. This is an area of research which has applications to cyber security, through cryptography, and to some aspects of coding theory. Graduate students supported by the award will receive training to contribute towards these projects.The primary goal of the Principal Investigator's project is to prove new relations between special cycles and automorphic forms. For example, the Principal Investigator will use recent advances in the theory of Borcherds products and integral models to prove the modularity of generating series formed from special cycles of arbitrary codimension on integral models of orthogonal Shimura varieties. The Principal Investigator will also prove new examples of generalized Gross-Zagier formulas, by expressing the intersection multiplicities of cycles on unitary Shimura varieties in terms of coefficients of integral kernels for automorphic L-functions.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.