自守L-函数、平移卷积和及相关数论问题

Many important problems in number theory are closely related to automorphic L-functions and shifted convolution sums. Moreover, there are deep relations between automorphic L-functions and shifted convolution sums. Based on the theory of automorphic forms, this project is intended to introduce new circle methods and study automorphic L-functions on GL(n), shifted convolution sums and related problems in number theory by applying various new methods and new tools such as the amplification method, the mollification method, the trace formula for GL(n), the Voronoi formula for GL(n) and estimation of high dimensional twisted character sums of Kloosterman's type. Specific studies include: (1) The subconvexity problem for automorphic L-functions based on new circle methods; (2) Special values of automorphic L-functions on GL(n) and its applications; (3) Mean values of shifted convolution sums based on new circle methods and its applications. This project is designed to provide new ideas and ways for the research in automorphic L-functions, shifted convolution sums and related problems in number theory.

数论中的许多重要问题都与自守L-函数和平移卷积和密切相关，并且自守L-函数与平移卷积和之间也有深刻的联系。本项目拟以自守形式理论为基础，探寻新型圆法，结合amplification方法、mollification方法、GL(n)的迹公式、GL(n)的Voronoi公式、高维Kloosterman型加权特征和的估计等新方法和新工具研究GL(n)的自守L-函数、平移卷积和及相关数论问题。具体研究内容包括：（1）基于新型圆法的自守L-函数的亚凸性问题研究；（2）GL(n)的自守L-函数的特殊值及其应用；（3）基于新型圆法的平移卷积和的均值估计及其应用。该研究旨在为自守L-函数、平移卷积和及相关数论问题的研究提供新的思路和途径。

自守形式理论是当代数论中的一个重要课题。本项目重点研究了自守L-函数的亚凸性、自守L-函数的特殊值、delta方法及其应用等方面的问题。具体说来，本项目在GL(2)、GL(3)和GL(3)×GL(2)的自守L-函数的亚凸性问题上取得新进展，发展了新型的delta方法，在GL(3)×GL(2)情形的Friedlander-Iwaniec猜想问题上取得新进展，研究了Dirichlet L-函数和加权Hecke-Maass L-函数的一类simultaneous non-vanishing问题等。研究成果分别发表在International Mathematics Research Notices、Publicacions Matemàtiques、Forum Mathematicum、The Quarterly Journal of Mathematics等国际期刊。