Special Cycles on Shimura Varieties and Derivative of L-Series

调查员主要从事三个项目的工作。第一个问题是Z上Hilbert模曲面中Hirzebruch-Zagier因子与算术CM圈之间的基本相交问题。我们的目标是证明J. Bruinier和研究者提出的一个漂亮的交集公式。它至少有两个应用程序。一个是著名的Chowla-Selberg公式的高度非平凡的推广，Colmez关于CM交换簇的Faltings高度的猜想。第二个是得到Igusa不变量CM值的分母的一个好的上界|完善劳特的一个猜想。第二个应用在Cohn-Lauter密码系统中也是非常重要的。第二个项目是与Bruinier的合作项目，他们试图研究扭曲Bocherds产品的CM值如何随着CM周期的变化而变化。他们还想找到CM值的因子分解公式。扭Borcherds积是一类系数在真实的二次域上的典型Hilbert模函数。第三个项目是解决退化Hilbert模曲面中的一个一般交问题，并利用它给出了Gross和Keating关于三个模对应的交的美丽公式的另一个证明。广泛影响：数学一直是科学和技术发展必不可少的。现在，数学对商业、金融、生物学和社会研究也变得越来越重要。数学的基础研究通常是“纯”数学（没有应用），但在意想不到的领域中越来越多地发现了意想不到的和深入的应用。例如，我研究的数论曾经被称为数学皇后，意思是它非常美丽，非常有吸引力，但在实践中毫无用处。而现在，它在编码理论和密码体制中占有极其重要的地位，这两者对我们的电子通信和电子商务交易以及国家安全都是至关重要的。算术和代数几何也是我研究的一部分，现在已经应用于人脸识别和经济学等工程领域。我所提出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理解，并将反过来为我们的社会福祉做出贡献。我提出的一个项目直接应用于密码系统，这对我们的国家安全和国民经济至关重要。

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 Z. 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, aconjecture 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 They try to study how the CM value of twisted Bocherds products behave as the CMcycle change. They also want to find a factorization formula for CM values. Twisted Borcherds products are a family of canonical Hilbert modular functions with coeficients 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.Broad Impact: Mathematics has always been essential to the development of science and technology. Now Mathematics has become increasingly important to business, finance, biology, and social studies too. Basic research in mathematics, which is usually `pure' mathematics (without application), has increasingly found unexpected and deep application in unexpected areas. For example, the number theory I studies used to call the queen ofMathematics, meaning that it is very beautiful and very attractive but useless in practice. But now it is extremely important in coding theory and cryptosystem, both are essentialto our electric communication and electronic business transaction as well as our national security. Arithmetic and algebraic geometry which is also in part of my proposed research has now applications in engineering such as face recognition and economics. My 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 our society's well-being. One of my proposed project has direct applications to cryptosystem,which is essential to our national security and national economy.