L-函数特殊值公式与算术应用

This project studies special value formulas of L-functions and its arithmetic applications. It contains mainly two parts:..(1) The relative trace formula approach for Gross-Zagier formula. The heights of Heegner points and the derivative central value of certain Rankin-Selberg L-functions are related via the Gross-Zagier formula. This formula is applied to Goldfeld’s solution of Gauss’ class number problem and Kolyvagin’s work on the BSD conjecture for elliptic curves with analytic rank less or equal to one. In 2012, Yuan-Zhang-Zhang obtained the most general Gross-Zagier formula using the theta lifting method. Meanwhile, a framework of using the relative trace formula to prove the Gross-Zagier formula is given by Tian-Yuan-Zhang. The advantage of the relative trace formula approach is illustrated by the recent work of Yun-Zhang on the Taylor expansion of L-functions on function fields. We hope our work, especially the study of archimedean smooth matching, can guide the future study of analogue question of higher rank case...(2) Explicit special value formula for triple L-functions. Similar to the case of Gross-Zagier formula, there is a conjecture between the heights of Gross-Kudla-Schoen cycles and the derivative central value of triple L-functions. This conjecture is proved by Ichino in the case of rank 0. Recently, using the p-adic version of such special value formulas, Darmon and his colleagues an important result on the equivalent BSD conjecture for elliptic curves over the rational field. On the other hand, explicit special value formula is important for actual arithmetic problems. The work of Ye Tian on the congruent number problem is an example of its application. We hope to obtain the similar explicit special value formula for triple L-functions as what we did for the Gross-Zagier formula. Moreover, We shall study its arithmetic application.

本课题是关于 L-函数特殊值公式及其算术应用的研究. 分为以下两个部分: (1) Gross-Zagier 公式的相对迹公式证明. Gross-Zagier 公式是一个将 Heegner 点的高度与某个 Rankin-Selberg L-函数的求导中心值联系起来的重要公式. 相对迹公式处理本类问题的优点已在近期 Yun-Zhang 关于函数域上L-函数的 Taylor 展开工作中体现出来. 希望通过我们的工作, 特别是对阿基米德情形下函数配对的研究, 给出未来高维情形下类似问题的指导. (2) 精确的 triple L-函数特殊值公式. 类比 Gross-Zagier 的情形, 存在猜想的公式联系 Gross-Kudla-Schoen cycle 的高度与某个triple L-函数求导中心值. 我们希望得到关于 triple L-函数的特殊值公式的精确化并进而研究其在算术问题上的应用.

本课题是关于 L-函数特殊值公式及其算术应用的研究. 主要包括三个方面: (1) Gross-Zagier 公式的相对迹公式证明. 通过克服阿基米德情形下算术光滑配对存在性这一技术难点, 我们通过相对迹公式的方法, 得到了最一般情形的 Gross-Zagier 公式. 我们的工作给出了高维类似问题的证明框架. (2) p-adic triple-L 函数的特殊值公式. 我们试图通过引入表示论的工具, 推广前人在模曲线情形下的此公式到任意 Shimura 曲线. 我们已在此方面取得一定进展. (3) 2 部分的 BSD 猜想. 通过研究椭圆函数L-函数的 modular symbol 的 norm compatible 性质, 我们得到了一族二次扭曲椭圆曲线的 BSD 猜想.