Topics in Arithmetic Geometry: Moduli Varieties, L-functions, Arakelov Theory and Their Interactions and Applications

这个项目属于数学的一个子领域，称为算术几何。该项目中的许多问题都是由这样一种哲学所激发的，即整数上的代数方程的信息可以通过几何方法获得。本项目所研究问题的解决将对密码学、理论物理和量子计算等领域的研究产生重大影响。研究者将研究模簇、L-函数、Arakelov理论及其相互作用。这项工作将有应用到解决的Birch和Swinnerton-Dyer猜想，其推广的Beilinson和Bloch，以及相关的问题。.特别是，将研究1）算术Gan-Gross-Prasad猜想; 2）具有任意长度的Drinfeld-Heegner圈的高度; 3）Gross-Zagier公式到高维的推广。

This project lies in a subfield of mathematics known as arithmetic geometry. Many of the questions in the project are motivated by the philosophy that information about algebraic equations over the integers can be obtained by geometric methods. Solution to the problems under study in this project will have substantial impact on research in cryptography, theoretical physics, and quantum computing.The investigator will study moduli varieties, L- functions, Arakelov theory and their interactions. This work will have applications to solution to the Birch and Swinnerton-Dyer conjecture, its generalization by Beilinson and Bloch, as well as related problems. . In particular, research will be done on 1) the arithmetic Gan-Gross-Prasad conjecture; 2) heights of Drinfeld-Heegner cycles with arbitrary length; 3) generalization of the Gross-Zagier formula to higher dimensions.