Arithmetic on Shimura Varieties and L-Series

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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.