CAREER: L-Functions and Subconvexity

本研究项目的重点是建立l -函数的性质，l -函数是复平面上的函数，编码各种数学结构的信息。l函数的一个例子是Riemann zeta函数，它对质数的信息进行编码，而质数对于计算机数据的安全传输至关重要。例如，其他类型的l函数可以帮助解释波在某些表面上传播的方式，这是物理学中感兴趣的话题。该项目的教育活动包括指导博士后研究员，培训研究生的研究，向本科生介绍研究，以及让高中生沉浸在强调研究中使用的思维类型的夏季数学课程中。本课题将重点研究l -函数理论中的一个重要而深刻的问题——次凸性问题。次凸性问题与等分布问题有关，涉及到l函数在其临界线上的非平凡上界的求取。这种边界在“导体掉落”的情况下尤其困难。这个项目的主要目标是为l函数建立新的子凸边界，并将现有的边界推向黄金标准（所谓的Weyl界）。该项目将考虑对称平方l函数（或这些函数的近似），l函数在“特殊点”表现出导体下降，以及l函数的强混合界，例如那些被狄利克雷字符扭曲的Hecke特征形式。方法将包括l -函数的矩族，互易公式，和自同态谱分析。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project is focused on establishing properties of L-functions, which are functions on the complex plane that encode information about various mathematical structures. An example of an L-function is the Riemann zeta function that encodes information about the prime numbers, which today are essential to the way computer data is securely transferred. Other types of L-functions can, for example, help explain the way waves propagate on certain surfaces, a topic of interest in physics. The educational activities of the project include mentoring a postdoctoral researcher, training graduate students in research, introducing undergraduates to research, and immersing high school students in a summer mathematics program that emphasizes the type of thinking used in research.This project will focus on investigating the subconvexity problem, an important and deep question in the theory of L-functions. The subconvexity problem is connected to equidistribution questions and involves obtaining non-trivial upper bounds for an L-function on its critical line. Such bounds are particularly difficult in “conductor-dropping’’ scenarios. The main goal of this project is to establish new subconvexity bounds for L-functions and push existing bounds towards the gold standard (the so-called Weyl bound). The project will consider the symmetric-square L-functions (or close approximations of these), L-functions at “special points” exhibiting conductor dropping, and strong hybrid bounds for L-functions, such as those of Hecke eigenforms twisted by Dirichlet characters. The methodology will include moments of L-functions in families, reciprocity formulae, and automorphic spectral analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.