数论是核心数学的主要分支之一。在这个领域，分析、代数、几何等核心数学学科高度交叉，被称作“算术代数几何”，对純数学的整体发展起着强烈的激励作用，因而备受数学界重视。特别是1960-70年代发展的朗兰兹纲领，包含自守形式等重要内容，被认为是数学史上最恢弘的研究计划之一。因此，深入研究自守形式理论，特别是考察迹公式在自守形式中的应用，建立起自守形式与一些具有算术意义的研究对象的关系，具有非常深刻的意义。. 本项目拟注重研究迹公式在自守形式中的应用，特别是在自守L-函数中的应用。注重利用Petersson迹公式及Kuznetsov迹公式，注重发展相对迹公式在自守形式中的最新关键技术，建立GL(n)上自守表示L-函数平均意义下的均值估计，建立GL(n)上自守L-函数与某些Periods积分的联系，进而研究自守L-函数的亚凸性界以及朗兰兹函子性猜想等相关问题。

Number theory is one of the basic research in the modern core mathematics. Analysis, algebra, geometry and other core disciplines of Mathematics are highlyoverlapping in this field. It is also called “arithmetic algebraic geometry”which promote the development of pure mathematics. Especially the Langlands program which was established up in 1960s-70s. The Langlands program contains thetheory of automorphic forms, automorphic representation and so on, and was regarded as one of the greatest research project. Hence studying the theory of auromorphic forms deeply, especially studying the application of trace formula to automorphic forms, establishing the connection between automorphic forms and some objects with arithmetic meaning, have deep significance.. To deal with this project, we plan to focus on the the application of trace formula to automorphic forms, especially to automorphic L-functions. We aim to study the Petersson Trace formula and the Kuznetsov trace formula, and develop the relative trace formula on GL(n). Throughout this project we plan to get some weighted mean values of the automorphic L-functions on GL(n), and found the connection between automorphic L-functions. Also, we could use this results to study the problems connected with the subconvexity bounds and the Langlands functoriality conjecture.