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.

该项目是由代数和数字理论计划在数学科学划分和刺激竞争研究（EPSCOR）的既定计划中共同资助的。该研究项目与数字理论基石的两个对象有关：L功能和自动形式。这些是非常特殊的功能类型，它包装了许多信息和对称性，因此是解决无数数学问题的关键。该项目将揭示L功能所拥有的新对称性，这反过来又对理解诸如素数之类的对象很有用。素数对于加密数字至关重要，该方法是将计算机数据牢固传输的方法。该项目还将揭示如何分发自动形式。这将部分回答算术量子混乱中的一些开放问题，这是数字理论和理论物理学界面的多学科领域。该项目将为研究生提供研究培训机会。首席研究人员还将继续参与一项外展计划，以准备贫困高中学生参加大学入口。在更详细的情况下，该项目的主要目标是：1）朝着随机波浪的猜想迈进，这预测自动态形式应该像随机波动一样行为像随机波和2）调查相互作用的配方对L型功能的相互作用。随机波浪的猜想可以用自动形式的力矩来表达，第二刻的陈述与熟悉的量子唯一的巨像问题相对应。该项目将通过调查自动形式的较高时刻，并对自动形式的分布有更深刻的了解，从而超越Que。互惠公式是L功能矩的某些精确公式，表现出令人惊讶的对称性。一个早期的例子是Motohashi的公式，它将Riemann Zeta函数的第四刻与自动形式的第三刻联系起来。该项目将发现互惠公式的新示例，该公式将阐明这种对称性的性质并产生新的应用，例如l功能的子范围界限。实现这两个目标的统一方法将是对自动形式和L功能的时刻进行研究，该方法是使用L功能分析理论和自动形态形式的频谱理论的方法。这项奖项反映了NSF的法定任务，并通过使用该基金会的智力优点和广泛的影响来评估NSF的法定任务。