CAREER: L-Functions and Subconvexity

本研究项目致力于建立L函数的性质，它是复平面上的函数，它编码关于各种数学结构的信息。L函数的一个例子是Riemann Zeta函数，它编码关于素数的信息，这些素数现在对于计算机数据安全传输的方式是必不可少的。例如，其他类型的L函数可以帮助解释波在某些表面上的传播方式，这是物理学中的一个感兴趣的话题。该项目的教育活动包括指导博士后研究人员，培训研究生的研究能力，介绍本科生进行研究，以及让高中生沉浸在一个强调研究中所用思维类型的暑期数学项目中。该项目将集中研究次凸性问题，这是L函数理论中一个重要而深入的问题。次凸性问题与均匀分布问题相联系，涉及到L函数在其临界线上的非平凡上界的获得。在“导线掉落”的情况下，这样的界限尤其困难。这个项目的主要目标是建立L函数的新的次凸性界限，并将现有的界限推向黄金标准(所谓的Weyl界)。这个项目将考虑对称平方的L函数(或这些函数的近似)，在“特殊点处”表现出导体脱落的L函数，以及L函数的强混合界，例如被狄里克莱特征符扭曲的黑克本征形式的界。该方法将包括L的时刻-家庭中的功能，互惠公式，和自同构谱分析。这个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智力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.