Arithmetic Volumes of Shimura Varieties

首席研究员将研究志村品种理论的各个方面。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。目标是在几何上定义的量（如体积和交叉多重）与来自其他数学分支（如表示理论）的量（没有先验几何意义）之间找到新的关系。除了在纯数学中的重要性外，志村变数在朗兰兹程序中也扮演着重要的角色，朗兰兹程序在理论物理学中越来越受到关注，志村变数是理解椭圆曲线和阿贝尔变数的重要工具，而椭圆曲线和阿贝尔变数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。志村变项研究的主要目标是证明其几何不变量与l函数之间的新关系。例如，首席研究员将利用Borcherds积理论和积分模型的最新进展，将酉型和正交型的Shimura变体的算术体积表示为l函数的特殊值。首席研究员还将证明广义Gross-Zagier公式的新例子，例如，通过用附在GSp的倒轴表示上的Langlands l函数的中心导数来表达嵌入西格尔三重曲线中的Shimura曲线的交叉多重(4)。类似的高维公式也被期待。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 Principal Investigator will study aspects of 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. The goal is to find new relations between geometrically defined quantities such as volumes and intersection multiplicities, and quantities that come from other branches of mathematics, like representation theory, that have no a priori geometric meaning. In addition to their importance in pure mathematics, Shimura varieties 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. The primary goal of the Principal Investigator's research on Shimura varieties is to prove new relations between their geometric invariants and L-functions. For example, the Principal Investigator will use recent advances in the theory of Borcherds products and integral models to express the arithmetic volumes of Shimura varieties of unitary and orthogonal type as special values of L-functions. The Principal Investigator will also prove new examples of generalized Gross-Zagier formulas, for example by expressing the intersection multiplicities of Shimura curves embedded into the Siegel threefold in terms of the central derivative of a Langlands L-function attached to a cuspidal representation of GSp(4). Similar higher dimensional formulas are also expected.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.