Arithmetic on Shimura Varieties and L-Series

该调查员主要从事以下项目。首先证明了酉型（n, 1）的Shimura变型上的CM环的Falltings高度（关于某个Arakelov除数）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将给出两种方法来从权值n+1的尖点形式构造Arakelov除数并证明它们是相等的。第二个项目是使用第一个项目的结果来证明酉型（n, 1）上的第一个Chow群的Gross-Zagier公式。第三个课题是更详细地理解第一个课题中出现的Rankin-Selberg积分。第四个项目是用我们在第二个项目中的方法证明Shimura曲线在全实数域上的Gross-Zagier-Zhang公式的变异体。第五个项目是研究CM阿贝尔品种的特殊自同态。其他项目包括算术函数的回拉，和格二曲线的计算。其中一些项目是联合项目。PI研究同一物体的两个不同方面之间的深层关系。对象是所谓的志村变量，一种特殊类型的多项式方程具有很多特殊的对称性。一方面，人们自然想知道它的算术，例如，在变量上的有理点和除数（一个额外的方程）。另一方面，也有l函数、爱森斯坦级数等各种解析对象漂浮在周围。此外，与Shimura变体相关的是群理论对象，例如自同构表示。PI调查了他们之间的深层关系。特别是，PI对著名的Gross-Zagier型公式的一些自然推广感兴趣，这对更受欢迎的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，这项研究在电信和密码学上也有潜在的非常重要的应用。

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.