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

算术流形、自守形式、指数和和 L 函数

基本信息

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

项目摘要

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.

这个项目的特点是在数学分析的界面上有两条研究线路，研究连续变化的研究，以及数论，研究源于整除概念的整数的性质。质数的精细分布，例如它们是否比其他数字更频繁地以某些数字块结束，被称为狄利克雷特字符的高度结构化的振荡序列捕捉到，这些字符产生了相关的L函数。本课题将证明关于L函数的某些特殊值和狄利克莱特征值的和的分析结果，并将具体研究由许多较小的因子组成的基础模对这些结果的影响以及对可用来证明它们的工具的影响。作为另一类待研究物体的原型，信号和运动(如光波或弦的振动)在被视为简单周期性运动的组合或叠加时往往更容易理解。类似于在其他空间上作为分析基础的波函数被称为本征函数，在从光谱几何到量子力学的各种学科中都是核心。这个项目将研究具有丰富对称性的空间上的快速振荡特征函数的行为，特别是它们的极值的显着性。本研究项目围绕两个主要主题，即高能特征函数在算术流形上的极值行为和解析数论中的深度和光滑方面。在某些具有特定几何和函数结构的算术流形上，联合特征函数(Hecke-Maass特征形式)表现出幂增长，这既不是通常预期的，也不是物理模型所预测的。PI将寻找极值增长并详细研究几类特定算术流形上的超范数和限制范数问题，以了解一般猜想和理解质量集中在算术流形上的现象，驱动它的精确结构，以及它在量子力学对应原理框架内的位置。在涉及特征标和大层次自同构形的数论问题中，深度和光滑性方面起着非常独特的作用，它们涉及到强大的或可因式分解的导体。本文将利用非阿基米德分析、解析数理论和谱理论研究强结构或可因式分解结构对L函数的非零性、次凸性、矩，以及涉及p-解析涨落的指数和的结构性影响。