Siegel 模形式是多个复变量自守函数最重要的一个例子, 它的理论具有很长的历史背景. 它和其它一些好的数论工具 (复分析, 代数几何) 一样, 一直在当代数论研究中处于非常重要的地位. L-函数在中心点处的取值问题在很多研究方向中都是很深刻的研究课题. 在证明 Siegel 模形式的 Fourier 系数的变号问题中 Koecher-Maass 级数起到了非常重要的作用. 在本项目中我们将研究 Koecher-Maass 级数在临界带型区域内的非零问题, 以及当亏格 n=2 时, 由特征型 F 生成的模函数的非零问题. 本项目的另一个研究工作是研究 Bocherer 猜想中的 twisted zeta 函数在中心点处的取值问题. 对这些问题的研究将使得 Siegel 模形式的理论体系更加丰富.

Siegel modular forms often are given as a first example of holomorphic modular forms in several variables. The theory meanwhile has a long traditional background and is a very important and active ares in modern research, combining in many nice ways of number theory, complex analysis and algebraic geometry. . Estimates of the values of L-functions at the central points are subject of intensive studies in various aspects. We will study the values of three kinds of Siegel modular function at the center points in our project. The Koecher-Maass series has significant applications in proving sign change results for the Fourier coefficients of Siegel cusp forms. First, we will study the nonvanishing problem for Koecher-Maass series attached to Siegel cusp forms of weight k and degree n in certain strips on the complex plane. Second, when the degree of F is two, we will study the nonvanishing problem of the modular function of F. At the end of this project, we will learn the values of the central points of the twisted zeta functions which were introduced by Bocherer Conjecture. The research of the above problems will make the theory of Siegel modular forms more richer.