Arithmetic on Shimura Varieties and Applications

点击翻译按钮获取中文摘要

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.