自守形式解析理论中的若干问题

The analytic theory of automorphic forms is an important part of modern number theory. Because this field is closely related to other branches of mathematics, the systematic introduction of new mathematical tools becomes possible. So that it is a very active field of mathematics. And at the same time, since it has a profound application in arithmetic problems, and makes an important role in the proof of some difficult problems, it gradually becomes the core area of modern number theory. With the Mobius randomness conjecture proposed by Sarnak (the Wolf Prize laureate) and the three main problems of automorphic L-functions in mind, this project plans to investigate sign changes of Fourier coefficients of automrophic forms on higher rank groups, the shifted convolution sums and the Mobius randomness conjecture related to automorphic forms, and their applications. To this end, based on our work on the theory of automorphic representations and analytic number theory, we design an approach different from traditional methods, which enables us to make some contributions to above problems. The applicant has obtained some interesting results in this field, and has some experiences in organizing scientific research activities. The applicant has been supported by Program for New Century Excellent Talents in University by the Ministry of Education, and Shandong Province Natural Science Foundation for Distinguished Young Scholars. The applicant won the Second Prize of State Natural Science Award in 2014 and the First Prize of Natural Sciences in University by the Ministry of Education in 2011.

自守形式的解析理论是现代数论的重要组成部分。由于这一领域与其它数学分支有着密切的联系，使得系统性地引入新的数学工具成为了可能，因而是极为活跃的数学领域。同时，由于其在数论问题中有着深刻的应用，在一些困难问题的解决中发挥了重要的作用，因而逐渐成为现代数论的核心领域。本项目拟围绕沃尔夫奖得主Sarnak提出的Mobius随机性猜想，以及自守L-函数的三大问题，开展高阶群上自守形式Fourier系数的变号问题、移位卷积和问题以及与之相关的Mobius随机性猜想的研究及其应用。为此，项目组基于在自守表示论和解析数论领域的工作基础，设计了有别于传统方法的研究途径，有望对上述问题做出贡献。申请人在这一领域得到了一些有意义的结果，具备一定的科研组织经验。曾先后入选“教育部新世纪优秀人才”,获“山东省杰出青年基金”，获2014年“国家自然科学二等奖”和“2011年教育部自然科学一等奖”。

自守形式的解析理论是现代数论的重要组成部分。由于这一领域与其它数学分支有着密切的联系，使得系统性地引入新的数学工具成为了可能，因而是极为活跃的数学领域。本项目拟围绕沃尔夫奖得主Sarnak提出的Mobius随机性猜想，以及自守L-函数的三大问题，开展高阶群上自守形式Fourier系数的变号问题、移位卷积和问题以及与之相关的Mobius随机性猜想的研究及其应用。按照上述研究计划，项目执行期间主要开展高阶群GL(m)自守形式解析理论的研究。通过将数论中的乘性函数理论和双重筛法等技术引入到了高阶自守形式领域，本项目在上述问题取得了系列进展。于Math. Ann., Math. Z., Forum Math., Canadian J. Math., Quart. J. Math.等刊物发表学术论文26篇。