Arithmetic Intersection, Modular Forms, and Complex Multiplication

研究者主要从事三个项目。第一个是整数上Hilbert模曲面上Hirzebruch-Zagier因子与算术CM环的基本相交问题。目标是证明J. Bruinier和研究者推测的一个漂亮的相交公式。它至少有两种应用。一个是对著名的Chowla-Selberg公式的高度非平凡的推广，这是Colmez对CM阿贝尔变种的Faltings高度的一个猜想。第二步是为Igusa不变量的CM值的分母获得一个很好的上界——改进Lauter的一个猜想。第二个应用在使用两条曲线的Cohn-Lauter密码系统中也具有重要的实际意义。第二个项目是与Bruinier合作的，他们试图研究扭曲Bocherds产品的CM值如何随着CM循环的变化而变化。题目还想找到CM值的因式分解公式。扭曲Borcherds积是实数二次域上带系数的正则希尔伯特模函数族。第三个课题是解决退化Hilbert模曲面上的一般相交问题，并用它再一次证明了Gross和Keating关于三个模对应的相交的美丽公式。本文的研究有助于对数论和算术几何中的一些算术和几何学科的基本和深入的理解，从而有助于社会的福祉。其中一个提议的项目直接应用于对国家安全和国民经济至关重要的密码系统。数论在编码理论和密码系统中变得越来越重要。算术和代数几何也是该研究的一部分，现在已应用于人脸识别和经济学等工程领域。

The investigator is mainly working on three projects. The first one is a fundamental and basic intersection problem between the Hirzebruch-Zagier divisors and arithmetic CM cycles in a Hilbert modular surface over integers. The goal is to prove a beautiful intersection formula conjectured by J. Bruinier and the investigator. It has at least two applications. One is a highly non-trivial generalization of the celebrated Chowla-Selberg formula, a conjecture of Colmez on Faltings' height of CM abelian varieties. The second is to obtain a nice upper bound for the denominator of the CM values of Igusa invariants---refining a conjecture of Lauter. The second application is also practically important in Cohn-Lauter cryptosystem using genus two curves. The second project is a joint one with Bruinier, in which they try to study how the CM value of twisted Bocherds products behave as the CM cycle change. They also want to find a factorization formula for CM values. Twisted Borcherds products are a family of canonical Hilbert modular functions with coefficients in the real quadratic field. The third project is to solve a general intersection problem in a degenerated Hilbert modular surface and use it to give another proof of the beautiful formula of Gross and Keating on intersection of three modular correspondences. The proposed research contributes to basic and deep understanding of some arithmetic and geometric subjects in number theory and arithmetic geometry, and will in turn contribute to the society's well-being. One of the proposed project has direct applications to cryptosystem, which is essential to national security and national economy. Number Theory is becoming extremely important in coding theory and cryptosystem. Arithmetic and algebraic geometry which is also in part of the proposed research has now applications in engineering such as face recognition and economics.