全实域上志村簇的特殊除子的算术Theta级数的模性定理

所谓Kudla纲领即志村簇上代数闭链、特殊除子和Eisenstein级数以及L-函数的特殊值或导数之间的相互关系的研究。最近Kudla纲领中志村簇上算术theta级数模性的证明以及算术基本引理的证明，极大地推动了纲领向着数论的核心问题迈进。本项目的主要研究目标在于推广模性定理至全实域的情形并证明这种情形下高维的类Gross-Zagier公式。项目的创新之处在于利用Arakelov几何中的格林流和光滑射影簇整模型上代数闭链的Beilinson-Bloch高度配对的一般猜想给出已有Kudla纲领的成果的几何解释，从而建立这种计算式的构造性的证明思路与这种公理化的存在性的理论之间的联系，相信这将会大幅拓展人们对于诸多数论问题包括算术基本引理、Rallis内积公式、BSD猜想、GGP猜想等的认识。

The so-called Kudla program is a series of studies on algebraic cycles, special divisors on Shimura varieties and their relation with Eisenstein series and special values or derivatives of L-functions. The most recent progress of Kulda program on the modularity of arithmetic theta series on Shimura varieties and the arithmetic fundamental lemma is leading its way to the core of number theory. The goal of the project includes the generalization of the modularity theorem to the total real field case and a proof of Gross-Zagier style formula in higher dimension in this case. The new idea of the project is to give a geometric interpretation of the known results of Kulda program by using Green currents in Arakelov geometry and the Belinson-Bloch height pairing conjecture on the integral model of smooth projective varieties. In this way, one can build the connection between the known constructive computational argument and the philosophical axiomatic theory. As a result, it can greatly make its contribution to the understanding and solutions to many number theory problems, such as arithmetic fundamental lemma, Rallis inner product formula, BSD conjecture and GGP conjecture.