Arithmetic on Shimura Varieties and Applications

数学对科学技术的发展一直是必不可少的。现在，数学对商业、金融、生物和社会科学也变得越来越重要。被认为是“纯数学”的数学基础研究，正越来越多地在意想不到的领域找到深入的应用。这个项目的研究领域就有这个味道。一方面，首席研究员正在研究志村变异，其定义本身涉及几何和表征理论。这些对象是现代数论的核心。在最简单的情况下，模曲线，它们已经被研究了一百多年，并且对证明费马大定理至关重要，费马大定理是数学中最著名和最古老的猜想之一。另一方面，研究者近年来的研究成果在代数曲线密码系统中也有一定的应用价值。这些对电子商务和通信的安全至关重要。研究者将继续研究这些类型的应用。在与Bruinier， Howard， Kudla和Rapoport合作的一个项目中，研究者旨在证明酉型的Shimura变体中的算术Chow循环的生成函数是模的。作为一个应用，他将证明更多的非阿贝尔情况的科尔梅兹猜想。在另一个项目中，研究者将尝试理解Colmez猜想的l函数侧，并获得如何攻击一般情况的线索。在另一个项目中，研究者将与合作者一起研究一些有趣的Borcherds项目的CM值。它们有两个目的：一个是证明一些经典猜想，另一个是在密码系统中的应用。

Mathematics has always been essential to the development of science and technology. Now mathematics is becoming increasingly important, too, to business, finance, biology, and social sciences. Basic research in mathematics, which is thought of as "pure" mathematics, is increasingly finding deep applications in unexpected areas. The research area of this project has this flavor. On the one hand, the principal investigator is investigating Shimura varieties, whose definition itself involves both geometry and representation theory. These objects lie at the center of modern number theory. In their simplest cases, the modular curves, they have been studied for more than one hundred years and were essential to the proof of Fermat's Last Theorem, one of the most famous and oldest conjectures in mathematics. On the other hand, the investigator's research results in recent years have become useful in algebraic curve cryptosystems. These are essential to the safety of electronic commerce and communication. The investigator will continue to work on these types of applications. In one of the projects joint with Bruinier, Howard, Kudla, and Rapoport, the investigator aims to prove that a generating function of arithmetic Chow cycles in the Shimura varieties of unitary type is modular. As an application, he will prove more non-Abelian cases of the Colmez Conjecture. In another project, the investigator will try to understand the L-function side of the Colmez conjecture and obtain clues on how to attack the general case. In yet another project, the investigator will study, with collaborators, CM values of some interesting Borcherds projects. These have two purposes: one is to prove some classical conjectures, and the other is an application to cryptosystems.