Siegel 模形式和两类L-函数的均值估计

Siegel modular forms were first introduced by Siegel in a paper of 1935 and nowadays often are given as a first example of holomorphic modular forms in several variables. The theory is a very important and active area in modern research, combining in many nice ways number theory, complex analysis and algebraic geometry.. Suppose the genus of Siegel modular form is 2, Andrianov proved that the Spinor zeta function has meromorphic continuation to the whole complex plane and satisfies a nice functional equation. So, it is very important to study this function. Suppose the genus of Siegel modular form is “n”, Koecher and Maass proved that the Koecher-Maass series has holomorphic continuation to the whole complex plane and satisfies a functional equation relating s→k-s,where k is the weight of the Siegel modular form. Because there are many nice analysis properties on these two functions, we could study the subconvexity bounds of these two functions. Similarly, we can consider the Riesz means of their coefficients. Moreover, we can estimate the higher power moments of the error term of their Riesz means and then establish the asymptotic formulas for the higher power moments.. The nonvanishing of certain L-functions is a crucial ingredient in the current development of generalized Ramanujian conjecture. Hence, in our last problem, we study the nonvanishing problem of these two L-functions at special points. We hope we can derive a simultaneous nonvanishing result for the central values of them. For the Spinor zeta function, the central point is s=k-1. We will study the first derivative of the Spinor zeta function, and hope there is an asymptotic formula. We will consider the similar problem on the Koecher-Maass series, but its central point is s=k/2.

1935年, Siegel在研究二次型的解析性质时第一次引进了Siegel 模形式的定义, 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.. 当Siegel 模形式的亏格是２时, Andrianov证明了它生成的spinor zeta 函数有很好的解析性质:它可以亚纯延拓到整个复平面上,并且有好的函数方程.当Siegel 模形式的亏格是ｎ时, Koecher 和 Maass证明了它生成的Koecher-Maass级数可以全纯延拓的整个复平面上, 并且有关于点 s=k/2对称的函数方程, 其中k是Siegel模形式的权. 基于这两类L-函数的解析性质,我们在本项目中重点研究这两类函数的凸性上界,Riesz均值,以及Riesz均值的高次均值估计.我们还将考虑这两个函数在中心点处是否非零,进而考虑它们的一阶导函数的中心点处的取值情况.

1935 年, Siegel 在研究二次型的解析性质时第一次引进了Siegel 模形式的定义, 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. . 当Siegel 模形式的亏格是２时, Andrianov 证明了它生成的spinor zeta 函数有很好的解析性质: 它可以亚纯延拓到整个复平面上,并且有好的函数方程. 基于这类L-函数的解析性质,在本项目中我们研究了它的的凸性上界, Riesz 均值. 在本项目中我们定义的 Riesz 均值是依赖于参数$\rho$ 的. 设F 是权为$ \kappa$亏格为2的Siegel 模形式，且当$\kappa $是偶数的时候F 是Maass子空间$S_\kappa^*(\Gamma_g)$ 的正交不补中的Siegel 模形式，当$1/2<\rho\leq 1$ 时，我们给出了Riesz 均值的Voronoi求和公式。进一步我们求出了Riesz 均值的二次均值的渐近公式, 这个结果是依赖于参数 $\rho$的. 当$\rho=1$ 时，我们求出这个Riesz 均值的高次均值的渐近公式. 目前我们所能做到的最高次数是5.在研究高次均值的时候我们主要使用的数学工具是Halasz-Montgomery 不等式，指数对理论等.. 在本项目的资助下，我们还研究了 m次对称幂L-函数的二次均值和高次均值. 同样，研究得到的结果也是依赖于参数 $\rho$ 的.