Arithmetic Intersection, Modular Forms, and Complex Multiplication

算术交集、模形式和复数乘法

• 批准号: 0555503
• 负责人: Tonghai Yang
• 金额: $ 18万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555503&HistoricalAwards=false
• 关键词: Arithmetic; Intersection; Modular; Forms; Complex

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.

调查员主要从事三个项目的工作。第一个是 基本和基本的交叉问题之间的 Hilbert空间中的Hirzebruch-Zagier因子与算术CM圈 整数上的模曲面目标是证明一个 J. Bruinier提出的一个美丽的交集公式， 调查员它至少有两个应用程序。一个是 对著名的 Chowla-Selberg公式，Colmez关于Faltings的一个猜想 CM阿贝尔品种的高度。二是 获得CM值的分母的一个很好的上限， Igusa不变量-完善劳特的一个猜想。第二 在Cohn-Lauter中，应用也具有重要的实际意义 一种使用亏格两条曲线的密码体制。 第二个项目是与Bruinier的联合项目， 其中，他们试图研究扭曲Bocherds产品的CM值如何表现为CM周期的变化。他们还想找到CM值的因子分解公式。 扭Borcherds积是一类系数为真实的的典型Hilbert模函数 二次域 第三个项目是解决一般的交叉问题， 退化Hilbert模曲面，并使用它来给出另一个 Gross和Keating的美丽公式的证明 三个模块对应的交集。 这项研究有助于基础和深入 理解一些算术和几何科目， 数论和算术几何，并将反过来 为社会的福祉做出贡献。一个拟议的 项目直接应用于密码系统， 对国家安全和国民经济至关重要。数字理论正在成为 在编码理论和密码系统中非常重要。算术和 代数几何，这也是拟议的研究的一部分， 现在已经应用于工程领域，如人脸识别， 经济学