高度与 L-函数

经典的Gross-Zagier公式将模曲线上Heegner点的高度与L-函数的导数的中心特殊值联系起来. 它对包括著名的 BSD 猜想在内的算术问题的研究起着极端重要的作用. 对 Gross-Zagier 的推广是当今世界数论领域的一大热点, 包括高阶群的推广,高阶求导的推广以及 p-adic Gross-Zagier 公式. 本项目拟在这三方面展开一系列的研究工作. 特别的, 对于高阶群的推广, 相对迹公式方法是目前公认最行之有效的一套方法. 这一方法中一大技术难点是所谓的阿基米德情形的算术光滑配对. 对此难点的研究国际上尚处在萌芽阶段, 目前唯一已知的结果是我们在 Shimura 曲线时所做的工作. 我们希望通过本项目, 将我们在曲线上的工作发展到高维情形, 使其真正成为一门研究代数闭链调和分析性质的理论.

The classical Gross-Zagier formula relates the heights of Heegner points and the central value of the derivative of L-functions. It is very important for arithmetical problems including the famous BSD conjecture. It is a hot issue in the present number theory field to generalize Gross-Zagier formula, including 1) Gross-Zagier formula for higher rank groups, 2) Gross-Zagier formula for higher derivatives; 3) p-adic Gross-Zagier formula. In this project, a series of research work will be carried out on these three aspects. Especially, the relative trace formula approach is believed as one of the most effective method for the problem 1). One of the major technical difficulties in this method is the so-called arithmetic smooth matching for archimedean places. The study of this problem is still in its infancy, and the only known result is the one we did for Shimura curves. We hope that through this project, we can develop our work on curves to high rank cases and make it really a theory for studying the harmonic analysis properties of algebraic cycles.