低权Jacobi形式及其应用

In the past decade, there are two remarkable breakthroughs in the applications of modular forms. The one is that a close relation of singular moduli with modular forms has been constructed, whick is provoked by the celebrated work of Borcherds, and the other is the mystery of mock theta functions left by Ramanujan has been uncovered. These two seemingly unrelated topics have been promoted and merged each other these years. Many new methods, new ideas and new concepts have emerged and a new system of theory is under way. The lead actors of the system are weak Maass forms, mock modular forms and Jacobi forms. The focuses of this program are Jacobi forms of low weight and their applications in singular moduli and mock theta functions. Specifically speaking, we will construct Jacobi forms of low weight by theta lifts and consider their arithmetic. The generating functions of singular moduli, mock theta functions and certain rank of partition functions will be constructed as holomorphic parts of real analytic Jacobi forms. We will also consider various correspondences among Jacobi forms,weak Maass forms and mock modular forms. This program involves many focuses of modern number theory, which are meeting ground for modern analysis, algebra and geometry.

近十年，在模形式理论的应用方面出现两大令人瞩目的突破。一个是在Borcherds杰出工作的推动下，奇异模理论取得极大进展；另一个是，关于Ramanujan遗留的mock theta函数的谜团已被揭开。这两方面看似无关，却相互促进，相互交织，有许多新方法、新思想、新概念不断涌现，一套新的理论体系初见端倪。居于该理论体系核心的有弱Maass形式、mock模形式和Jacobi形式。本项目主要以低权Jacobi形式为核心，考虑低权Jacobi形式的算术性质及其在奇异模和mock theta函数方面的应用。具体讲，构造低权Jacobi形式,考虑其算术性质，进而利用低权Jacobi形式性质来研究奇异模、mock theta函数和分拆函数，以及与弱Maass形式、mock模形式的各种关联。本课题涉及当代数论研究的核心热点问题，是当代数论和分析、代数、几何的交叉领域，有重要的研究意义和价值。

本项目在Jacobi形式的算术理论以及堆垒素数论的一些课题做了比较深入的研究，取得了比较好的进展。低权Jacobi形式与新兴的调和Maass形式，mock模形式有密切的关系，在椭圆曲线、组合数论、顶点算子代数及数学物理方面都有广泛的应用。具体来讲，我们做了如下几个方面的工作：1. 深入研究了非全纯的skew-holomorphic Jacobi形式的算术性质，包括它的迹公式和维数公式，并且对skew-holomorphic Jacobi理论进行了全面系统的发展；2. 考虑了低权Jacobi形式与四元代数的算术性质，推广了Gross的一个结果，给出Eichler-Selberg迹公式一个新的证明；3. 考虑了几类mock theta 函数的一些性质；4. 研究了某些混合幂的Waring—Goldbach问题及例外集问题，还证明了满足一定条件的大偶数可表示为四个素数平方与117个2的方幂之和；5.还考虑了素变量丢番图逼近,得到更优的上界。