Arithmetic on Shimura Varieties and L-Series

志村品种和 L 系列的算术

• 批准号: 1200380
• 负责人: Tonghai Yang
• 金额: $ 25.5万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200380&HistoricalAwards=false
• 关键词: Arithmetic; Shimura; Varieties; Series

The investigator is mainly working on following projects. The first is to prove a precise relation between the Falltings' height of a CM cycle (with respect to certain Arakelov divisor) on a Shimura variety of unitary type (n, 1) and the central derivative of certain Rankin-Selberg L-function. Along the way, the PI will give two ways to construct the Arakelov divisor from a cusp form of weight n+1 and prove them to be equal. The second project is to use the result in the first project to prove a Gross-Zagier formula for the first Chow group over the Shimura variety of unitary type (n, 1). The third project is to understand the Rankin-Selberg integral appeared in the first project in more detail. The fourth project is to prove variants of Gross-Zagier-Zhang formula over a totally real number field for Shimura curve using our method in the second project. The fifth project is to study special endomorphisms of CM Abelian varieties. Other projects include pull-back of arithmetic theta functions, and genus two curve computations. Some of these projects are joint projects.The PI investigates the deep relation between two different aspects of the same object. The object is the so-called Shimura variety, a special type of polynomial equations with a lot of special symmetries. On the one side, one would like to know naturally its arithmetics, e.g., rational points and divisors (one extra equation) on the varieties. On the other hand, there are also various analytic objects such as L-functions and Eisenstein series floating around. In addition, associated to the Shimura varieties are group theoretic objects such that automorphic representations. The PI investigates the deep relations among them. In particular, the PI is interested in some natural generalization of the well-known Gross-Zagier type of formulas, which has very significant implication to the even more popular Birch and Swinnerton-Dyer conjecture (one of the million dollar problems). In addition to its importance in mathematics, this research has also potentially very important application to telecommunication and cryptography.

研究者主要从事以下项目。首先证明了酉型（n，1）Shimura簇上CM圈（关于某个Arakelov因子）的Falltings高度与某个Rankin-Selberg L-函数的中心导数之间的精确关系。沿着的方式，PI将给出两种方法来构造阿拉克洛夫因子从尖形式的重量n+1，并证明他们是平等的。第二个项目是利用第一个项目中的结果证明酉型（n，1）的Shimura簇上第一Chow群的Gross-Zagier公式。第三个项目是更详细地理解第一个项目中出现的Rankin-Selberg积分。 第四个项目是证明变体 在第二个项目中，我们用我们的方法研究了全真实的数域上的Gross-Zagier-Zhang公式。 第五个项目是研究CM阿贝尔簇的特殊自同态。其他项目 包括 算术theta函数的拉回，以及两个曲线计算的亏格。其中一些项目是联合项目。PI调查同一对象的两个不同方面之间的深层关系。对象是所谓的志村簇，一种特殊类型的多项式方程，具有许多特殊的对称性。一方面，人们想自然地知道它的算术，例如，有理点和除数（一个额外的方程）的品种。另一方面，也有各种各样的分析对象，如L-函数和爱森斯坦级数。此外，与志村变种相关的是群论对象，例如自守表示。PI研究了它们之间的深层关系。特别是，PI对着名的Gross-Zagier类型公式的一些自然推广感兴趣，这对更流行的Birch和Swinnerton-Dyer猜想（百万美元问题之一）具有非常重要的意义。除了在数学上的重要性，这项研究也有潜在的非常重要的应用电信和密码学。