本项目拟研究Petersson迹公式和Kuznetsov迹公式，进而研究Maass尖形式、全纯Hilbert尖形式和Hilbert-Maass尖形式的解析性质．申请人已经在迹公式及其应用方面有了一定的工作基础．在本项目实施过程中，我们希望得到不加权的Petersson迹公式和Kuznetsov迹公式，进而研究如下四个问题：.(1)Maass尖形式对应的对称方幂L-函数在特殊点的值的分布问题；.(2)全纯Hilbert尖形式对应的对称方幂L-函数的密度定理及其应用;.(3)全纯Hilbert尖形式的Elliott-Montgomery-Vaughan类型的大筛法型不等式及其应用;.(4)Hilbert-Maass尖形式的“垂直版”Sato-Tate猜想和它的收敛速度．.本项目预期的研究结果将会对自守形式的Hecke特征值的分布问题和自守L-函数在特殊点的值的分布问题产生重要影响．

In this project, we focus on Petersson trace formula and Kuznetsov formula and apply them to study the analytic properties of Maass cusp forms, holomorphic Hilbert cusp forms and Hilbert-Maass cusp forms. The applicant has a strong background in the theory of trace formulas and their applications. During the project period, we plan to obtain unweighted Petersson trace formula and unweighted Kuznetsov formula and then apply them to study the following four problems:.(1) Distribution of values of symmetric power L-functions of Maass cusp forms at one;.(2) A density theorem on symmetric power L-functions of holomorphic Hilbert cusp forms and some applications;.(3) A large sieve inequality of Elliott-Montgomery-Vaughan type for holomorphic Hilbert cusp forms with applications to Linnik’s problem;.(4) Quantitative version of Sato-Tate vertical distribution of the Satake parameter of Hilbert-Maass cusp forms..The expected results of this project will have important effects on the study of distribution of Hecke eigenvalues of automorphic forms and distribution of values of automorphic L-functions at one.