Borcherds提升理论是目前研究数论及其相关领域的重要工具之一，并在其中有着非常重要且广泛的应用。在本项目中，我们主要利用Borcherds提升理论研究以下三个与数论相关的课题：（1）Gross-Zagier判别式公式及其推广；（2）关于小CM点的Gross-Zagier猜想；（3）广义Weber函数的Fourier系数的相互关系。通过对这三个课题的研究，本项目将帮助我们进一步了解虚二次域环类多项式的性质、高权自守Green函数的小CM值与代数数的关系以及广义Weber函数的性质，并期望这些结果有助于其他相关课题的研究，例如Gross-Zagier猜想对于大CM点的情况、著名的Yui-Zagier猜想、Dedekind eta函数的Fourier系数的性质等。

The theory of Borcherds lifts is one of the key tools in the study of number theory and its related areas and has a variety of important applications in them. In this project, we employ the theory of Borcherds lifts to study the following three topics in number theory: (1) the Gross-Zagier discriminant formula and its extensions; (2) the Gross-Zagier conjecture on small CM points; and (3) interrelations between the Fourier coefficients of the generalized Weber functions. Through exploring these three topics, this project will help us understand deeper and more about the properties of certain ring class fields associated with imaginary quadratic fields, the relationship between the small CM values of higher weight automorphic Green functions and algebraic numbers, and the properties of the generalized Weber functions, which are expected to be helpful to other subsequent studies such as the Gross-Zagier conjecture on big CM points, the Yui-Zagier conjecture, the properties of the Fourier coefficients of the Dedekind eta function, etc.