Higher Codimension Cycles on Shimura Varieties

该项目的主要重点是志村品种的理论。这些是特殊类型的高维曲面，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。除了它们在纯数学中的重要性外，它们在朗兰兹纲领中也起着至关重要的作用，朗兰兹纲领在理论物理学中越来越受到关注，并且是理解椭圆曲线和阿贝尔簇的重要工具，现在椭圆曲线和阿贝尔簇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。该项目的主要研究者的主要目标是证明特殊循环和自守形式之间的新关系。例如，首席研究员将使用Borcherds产品和积分模型理论的最新进展来证明由正交Shimura品种的积分模型上的任意余维的特殊循环形成的生成系列的模块性。首席研究员还将证明广义Gross-Zagier公式的新例子，通过表示在自守L-函数的积分核系数的酉志村品种的循环的交叉多重性。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.