Arithmetic Manifolds, Automorphic Forms, Exponential Sums, and L-Functions

算术流形、自守形式、指数和和 L 函数

• 批准号: 1503629
• 负责人: Djordje Milicevic
• 金额: $ 13.5万
• 依托单位: Bryn Mawr College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1503629&HistoricalAwards=false
• 关键词: Arithmetic; Manifolds; Automorphic; Forms; Exponential

This project features two lines of research at the interface of mathematical analysis, the study of continuous change, and number theory, the study of properties of integers borne out of the notion of divisibility. The finer distribution of prime numbers, such as whether they end in certain blocks of digits more often than others, is captured by highly structured oscillating sequences known as Dirichlet characters, which give rise to the associated L-functions. This project will prove analytic results about certain special values of L-functions and sums of the values of Dirichlet characters, and it will specifically investigate the impact of the underlying modulus being composed of many smaller factors on these results and on the tools available to prove them. As a prototype of the other class of objects to be studied, signals and motions (such as light waves or vibrations of a string) are often much better understood when viewed as combinations, or superpositions, of simple periodic motions. The wave-like functions that analogously serve as building blocks of analysis on other spaces are known as eigenfunctions and are central in disciplines ranging from spectral geometry to quantum mechanics. This project will investigate the behavior of rapidly oscillating eigenfunctions on spaces with a rich set of symmetries that are arithmetic in nature, in particular how pronounced are their extreme values.This research project centers around two principal themes, that of extremal behavior of high-energy eigenfunctions on arithmetic manifolds and that of the depth and smooth aspects in analytic number theory. On certain arithmetic manifolds with a specific geometric and functorial structure, the joint eigenfunctions (Hecke--Maass eigenforms) exhibit power growth, which is neither generically expected nor predicted by physical models. The PI will seek out extremal growth and investigate in detail the sup-norm and restriction norm problems on several specific classes of arithmetic manifolds to inform general conjectures and understanding of the phenomenon of concentration of mass on arithmetic manifolds, the precise structure that drives it, and its place within the framework of the correspondence principle of quantum mechanics. In number-theoretic problems involving characters and automorphic forms of large level, the depth and smooth aspects, which are concerned with highly powerful or factorable conductors, play a very distinctive role. The structural impact of the powerful or factorable structure on nonvanishing, subconvexity, and moments of L-functions, as well as exponential sums involving p-adically analytic fluctuations will be studied using non-archimedean analysis, analytic number theory, and spectral theory.

该项目在数学分析的界面，持续变化的研究和数量理论的界面上进行了两条研究，这是对整数属性的研究。质数的分布比其他数字更频繁地在某些数字块中结束，被称为dirichlet字符的高度结构化振荡序列捕获，这引起了相关的L功能。该项目将证明有关dirichlet字符值的某些特殊值的特殊值和总和的分析结果，它将专门研究由许多较小因素组成的基础模量对这些结果以及可用的工具的影响。作为要研究的其他类别对象的原型，当将简单周期性运动视为组合或叠加时，通常可以更好地理解信号和动作（例如弦的光波或振动）。类似的波动函数将其他空间上的分析构建块称为特征函数，并且在从光谱几何学到量子力学的学科中是中心的。该项目将调查在本质上具有丰富对称性的空间上快速振荡征函数的行为，尤其是它们的极端价值。该研究项目围绕两个主要主题，即对高度能力特征的极端行为，这些主题是在算术流上的高度能量和平滑等方面的高度和平稳的方面。在具有特定几何和功能结构的某些算术歧管上，关节特征函数（Hecke-maass eigenforms）表现出功率增长，这既不是通常的，也不是物理模型预测的。 PI将寻求极大的增长，并详细研究几种特定类别的算术流形的SUP-NORM和限制规范问题，以告知一般猜想和对质量集中浓度的现象，对算术歧管的浓度，使其驱动量的精确结构及其在量子力学相应原理的框架内的位置。在涉及角色和大型自动形式的数量理论问题中，与高功能或可分解的导体有关的深度和光滑方面起着非常独特的作用。强大或可分解结构对L功能的非变化，亚凸度和力矩的结构影响以及涉及P-AFAIPAINE分析波动的指数总和将使用非建筑分析，分析数理论和光谱理论来研究。

项目成果

The Second Moment Theory of Families of 𝐿-Functions–The Case of Twisted Hecke 𝐿-Functions

• DOI: 10.1090/memo/1394
• 发表时间: 2023-02
• 期刊: Memoirs of the American Mathematical Society
• 影响因子: 1.9
• 作者: V. Blomer;É. Fouvry;E. Kowalski;P. Michel;Djordje Milićević;W. Sawin