L-函数导数的研究是当代数论前沿问题之一，它与许多核心问题密切相关。本项目主要研Gross-Zagier公式和Kudla纲领，它们是研究L-函数一阶导数算术几何表示的两个重要方法。我们将在更广泛的意义下研究二者，具体研究内容包括：（1）在我们前期工作的基础上(算术Siegel-Weil公式)，研究模曲线上的算术内积公式,并且揭示它与经典Gross-Zagier公式关系（已经得到部分结果）；2）研究一般Shimura簇上的Kudla纲领，得到更一般的算术Siegel-Weil公式和算术内积公式。并且应用袁-张-张的方法，寻找Gross-Zagier公式和Kudla纲领的深刻联系；（3）在函数域上建立和发展Kudla纲领，并且结合恽之伟和张伟在函数域上的工作，促进L-函数算术几何性质的研究。

The derivative of L-functions is one of the most important issues in modern number theory, and it has deep relations with some core problems. Gross-Zagier formula and Kudla program are two important methods to study the arithmetic geometry of the first derivatives of L-functions. We will study both of them in more broader sense, including the following three topics:(1)Based on our previous works (arithmetic Siegel-Weil formula), we will prove the arithmetic inner product formula on modular curve and relate it to the classical Gross-Zagier fromula (We have obtained part of the result) . (2)We will study the general Shimura varieties and prove the arithmetic Siegel-Weil formula and inner product formula. Thus, we could find the deep relations between Gross-Zagier formula and Kudla program by Yuan-Zhang-Zhang's methods. (3)We will develop the Kudla program on function fields, and relate it to the work of Zhiwei Yun and Wei Zhang's. It will promote the study of the arithmetic geometry of the L-function.