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

该项目的特点是在数学分析的界面上进行两条研究线，研究连续变化，以及数论，研究整除性概念中整数的性质。素数的更精细分布，例如它们是否比其他数字更经常地在某些数字块中结束，被称为狄利克雷特征的高度结构化振荡序列捕获，这产生了相关的L函数。本项目将证明关于L-函数的某些特殊值和Dirichlet特征标值之和的分析结果，并将具体研究由许多较小因子组成的基础模对这些结果的影响以及可用于证明它们的工具。作为另一类待研究对象的原型，信号和运动（如光波或弦的振动）通常在被视为简单周期运动的组合或叠加时更容易理解。类似地作为其他空间分析的基石的波函数被称为本征函数，并且是从光谱几何到量子力学的学科的核心。本研究课题主要研究具有丰富的算术对称性的空间上的本征函数的快速振荡行为，特别是其极值的显著性。本研究课题主要围绕算术流形上的高能本征函数的极值行为和解析数论中的深度和光滑性两个主题展开。在具有特定几何和函子结构的算术流形上，联合本征函数（Hecke-Maass本征形）表现出幂增长，这既不是一般预期的，也不是物理模型预测的。PI将寻找极端增长并详细研究几种特定类别算术流形上的超规范和限制规范问题，以提供一般猜想和对算术流形上质量集中现象、驱动它的精确结构及其位置的理解。它在量子力学对应原理框架内的位置。在涉及大能级的特征标和自守形式的数论问题中，与高幂次或可因式分解导体有关的深度和光滑方面起着非常独特的作用。我们将使用非阿基米德分析、解析数论和谱理论来研究强大的或可因式分解的结构对L-函数的非零性、次凸性和矩以及涉及p-基解析波动的指数和的结构影响。

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.