Analytic Theory of Automorphic Forms and L-Functions

该项目由数学科学部的代数和数论计划以及刺激竞争研究的既定计划（EPSCoR）共同资助。这个研究项目涉及数论的两个基石：L-函数和自守形式。这些都是非常特殊的函数类型，它们包含了大量的信息和对称性，因此是解决无数数学问题的关键。该项目将揭示L函数所拥有的新对称性，这反过来将有助于理解素数等对象。素数在密码学中是必不可少的，密码学是一种安全传输计算机数据的方法。该项目还将揭示自守形式是如何分布的。这将部分回答算术量子混沌中的一些开放性问题，这是数论和理论物理接口的多学科领域。该项目将为研究生提供研究培训机会。主要研究者还将继续参与帮助贫困高中生准备大学入学的外联计划。更详细地说，该项目的主要目标是：1）在随机波猜想方面取得进展，该猜想预测自守形式的行为应该像随机波; 2）研究L函数的互易公式类。随机波猜想可以用自守形式的矩来表述，其中二阶矩的陈述对应于熟悉的量子唯一遍历问题。这个项目将超越QUE，通过研究自守形式的高阶矩，并对自守形式的分布有更好的理解。倒易公式是L-函数矩的精确公式，具有令人惊讶的对称性。一个早期的例子是Motohashi公式，它将黎曼Zeta函数的四阶矩与自守形式的三阶矩联系起来。这个项目将发现互易公式的新例子，这将揭示这种对称性的本质，并产生新的应用，例如L函数的次凸性边界。这两个目标的统一方法将是自守形式和L函数的矩的研究，使用L函数的分析理论和自守形式的谱理论的方法。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。

This project is jointly funded by the Algebra and Number Theory program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR). This research project is concerned with two objects at the cornerstones of number theory: L-functions and automorphic forms. These are very special types of functions, which package a lot of information and symmetry, and for this reason are the key to solving a myriad of mathematical problems. The project will reveal new symmetries possessed by L-functions, which in turn will be useful for understanding objects such as the prime numbers. Prime numbers are essential in cryptography, a method by which computer data is securely transferred. The project will also reveal how automorphic forms are distributed. This will partially answer some open questions in Arithmetic Quantum Chaos, a multidisciplinary field at the interface of number theory and theoretical physics. The project will provide research training opportunities for graduate students. The principal investigator will also continue to be involved in an outreach program to prepare underprivileged high school students for college entrance.In more detail, the principal goals of this project are to: 1) Make progress towards the Random Wave Conjecture, which predicts that automorphic forms should behave like random waves and 2) investigate the class of reciprocity formulae for L-functions. The Random Wave Conjecture can be formulated in terms of moments of automorphic forms, with the statement for the second moment corresponding to the familiar Quantum Unique Ergodicity problem. This project will go beyond QUE, by investigating higher moments of automorphic forms, and yielding a finer understanding of the distribution of automorphic forms. The reciprocity formulae are certain exact formulae for moments of L-functions, exhibiting some surprising symmetry. An early example is Motohashi's formula, which relates the fourth moment of the Riemann Zeta function to the third moment of automorphic forms. This project will discover new examples of reciprocity formulae, which will shed light on the nature of such symmetries and yield new applications, such as subconvexity bounds for L-functions. The unifying approach for both goals will be a study of moments of automorphic forms and L-functions, using methods from the analytic theory of L-functions and the spectral theory of automorphic forms.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.