Analytic Theory of Automorphic Forms and L-Functions

自守形式和 L 函数的解析理论

• 批准号: 2344044
• 负责人: Rizwanur Khan
• 金额: $ 16.73万
• 依托单位: University of Texas at Dallas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2344044&HistoricalAwards=false
• 关键词: Analytic; Theory; Automorphic; Forms; Functions

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.

该项目由数学科学部代数和数论项目以及刺激竞争性研究既定项目（EPSCoR）共同资助。该研究项目涉及数论基石的两个对象：L 函数和自守形式。这些是非常特殊的函数类型，它们包含大量信息和对称性，因此是解决无数数学问题的关键。该项目将揭示 L 函数所具有的新对称性，这反过来将有助于理解素数等对象。质数在密码学中至关重要，密码学是一种安全传输计算机数据的方法。该项目还将揭示自同构形式是如何分布的。这将部分回答算术量子混沌中的一些悬而未决的问题，算术量子混沌是数论和理论物理交叉领域的一个多学科领域。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一项外展计划，为贫困高中生进入大学做好准备。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，该猜想预测自同构形式应表现得像随机波，并且 2) 研究 L 函数的倒易公式类别。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性问题。该项目将超越 QUE，通过研究自同构形式的更高矩，并更好地理解自同构形式的分布。倒易公式是 L 函数矩的某些精确公式，表现出一些令人惊讶的对称性。一个早期的例子是本桥公式，它将黎曼 Zeta 函数的四阶矩与自守形式的三阶矩联系起来。该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 函数的次凸界。这两个目标的统一方法将是使用 L 函数解析理论和自同构形式谱理论的方法来研究自同构形式和 L 函数的矩。该奖项反映了 NSF 的法定使命，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来提供支持。