Arithmetic on Shimura Varieties and L-Series

调查员主要从事以下项目的工作。首先证明了单位型(n，1)Shimura簇上CM圈的Fallings高度(关于某个Arakelov因子)与某个Rankin-Selberg L函数的中心导数之间的精确关系。在此过程中，PI将给出由权n+1的尖点形式构造Arakelov除数的两种方法，并证明它们是相等的。第二个项目是利用第一个项目的结果证明了第一个Chow群在酉型(n，1)的Shimura簇上的一个Gross-Zagier公式。第三个项目是更详细地了解第一个项目中出现的Rankin-Selberg积分。第四个项目是在第二个项目中用我们的方法证明了Shimura曲线的Gross-Zagier-Zhang公式在全实数域上的变体。第五个项目是研究CM-Abel变种的特殊自同态。其他项目包括算术theta函数的回拉，以及亏格2曲线计算。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。对象是所谓的下村变种，这是一种具有许多特殊对称性的特殊类型的多项式方程。一方面，人们想自然地知道它的算术，例如，关于簇的有理点和因子(一个额外的方程)。另一方面，也有各种各样的分析对象，如L函数和艾森斯坦级数。此外，与Shimura变种相关的是群论对象，使其具有自同构表示。PI考察了它们之间的深层次关系。特别是，PI对著名的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外，这项研究还可能在电信和密码学方面有非常重要的应用。

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.