Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity

等分布、自守形式的周期积分和次凸性

• 批准号: 2302079
• 负责人: Peter Humphries
• 金额: $ 15.79万
• 依托单位: University of Virginia Main Campus
• 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: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302079&HistoricalAwards=false
• 关键词: Equidistribution; Period; Integrals; Automorphic; Forms

This project will investigate some open questions in arithmetic quantum chaos, a multidisciplinary field at the interface of number theory and theoretical physics. The mathematical study of quantum chaos involves the interplay between quantum mechanics and classical mechanics. The goal is to understand how chaotic behavior of the latter, such as ergodic billiard flow, is reflected by chaotic behavior of the former. One of the main tasks of this project is to investigate these relations in settings where there are additional ties to number theory: the study of prime numbers and their distribution. The PI will also supervise undergraduate and high school students as part of the UVA number theory REU, as well as continue to train PhD students.The PI will investigate problems in the analytic theory of automorphic forms that involve applications of subconvex bounds for L-functions. One major problem to be resolved is restricted arithmetic quantum unique ergodicity, which involves the study of the mass equidistribution of automorphic forms on curves on the modular surface. The PI will additionally prove new subconvex bounds for Rankin-Selberg L-functions. The methods to prove these results will rely on new instances of spectral reciprocity, which are identities for moments of L-functions, and will be proven using period formulae relating integrals of automorphic forms and their L-functions.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.

该项目将在数字理论和理论物理学界面上的多学科领域中调查一些算术量子混乱中的一些开放问题。量子混乱的数学研究涉及量子力学与经典力学之间的相互作用。目的是了解后者的混乱行为是如何通过前者的混乱行为反映出的。该项目的主要任务之一是在与数字理论有其他联系的环境中调查这些关系：素数及其分布的研究。 PI还将监督本科生和高中生作为UVA编号理论REU的一部分，并继续培训博士学位的学生。PI将调查涉及L-Functions子凸界的自动形式分析理论中的问题。要解决的一个主要问题是限制算术量子独特的牙齿性，这涉及对模块化表面曲线上自动形式的质量等分分配。 PI还将证明Rankin-Selberg L功能的新子凸界。证明这些结果的方法将依赖于光谱互惠的新实例，这些实例是L功能时刻的身份，并将使用与自动形式的积分及其L功能相关的时期公式进行证明，这奖反映了NSF的法定任务，并通过使用基础的智力效果和广泛的范围来评估支持，并通过评估值得评估。