Analytic Theory of Automorphic Forms and L-Functions

该项目由数学科学部的代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。本研究计划关注数论的两个基石：l -函数和自同构形式。这些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函数所具有的新的对称性，这反过来将有助于理解诸如素数之类的对象。素数在密码学中是必不可少的，这是一种安全传输计算机数据的方法。该项目还将揭示自同构形式的分布方式。这将部分回答算术量子混沌中的一些开放问题，这是数论和理论物理的一个多学科领域。该项目将为研究生提供研究培训机会。首席研究员还将继续参与帮助贫困高中学生进入大学的外展计划。更详细地说，这个项目的主要目标是：1)在随机波猜想方面取得进展，该猜想预测自同构形式应该表现得像随机波；2)研究l函数的互易公式类。随机波猜想可以用自同构形式的矩来表示，第二个矩的表述对应于我们熟悉的量子唯一遍历性问题。这个项目将超越QUE，通过研究自同构形式的更高矩，并产生对自同构形式分布的更好理解。互易公式是l -函数矩的精确公式，具有惊人的对称性。一个早期的例子是Motohashi的公式，它将Riemann Zeta函数的第四阶矩与自同构形式的第三阶矩联系起来。该项目将发现互易公式的新例子，这将阐明这种对称性的本质并产生新的应用，例如l函数的次凸边界。这两个目标的统一方法将是研究自同构形式和l函数的矩，使用l函数的解析理论和自同构形式的谱理论的方法。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 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.