L函数中心值与椭圆曲线的算术

椭圆曲线的有理点问题具有重大的理论和实用价值。L函数及其导数的中心值蕴涵了丰富的算术几何信息，与椭圆曲线有理点问题密切相关。本项目希望通过相对迹公式等自守表示论和解析数论中的方法来研究L函数及其导数的中心值，进一步诠释其算术几何意义；并借助BSD猜想方面的最新研究成果，来研究椭圆曲线有理点问题，如Ш群阶数分布等。特别是研究高斯类数问题在椭圆曲线情形的类比，能够给相关领域带来新的理解和视角。

The study of rational points on elliptic curves is both theoretically and practically significant. The central values of L-functions and their derivatives carry rich arithmetic geometric information, and are closely related to the rational points on elliptic curves. The goal of this project is to study the central values of L-functions and their derivatives, using tools such as the relative trace formulas from automorphic representation theory and analytic number theory; and to further understand the rational points on elliptic curves, with the help from the recent progress on the study of the BSD conjecture. Especially, we will study an analog of Gauss' class number problem in the case of elliptic curves. We hope this project will bring new knowledge and approaches to the study of elliptic curves.