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

这个项目的特点是在数学分析的界面上进行两方面的研究，连续变化的研究和数论，研究由可整除性概念产生的整数的性质。质数的精细分布，例如它们是否比其他数字更经常地以某些数字块结尾，是由称为狄利克雷字符的高度结构化振荡序列捕获的，它产生了相关的l函数。本项目将证明关于l函数的某些特殊值和狄利克雷特征值和的解析结果，并将具体研究由许多较小因子组成的底层模对这些结果的影响以及对可用的证明工具的影响。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）当被视为简单周期运动的组合或叠加时，通常会更好地理解。类似于其他空间分析的构建块的波状函数被称为特征函数，是从光谱几何到量子力学等学科的中心。本项目将研究快速振荡特征函数在具有丰富对称性的空间上的行为，这些对称性本质上是算术的，特别是它们的极值有多明显。本研究项目围绕两个主要主题展开，即算术流形上高能特征函数的极值行为和解析数论中深度和光滑方面的研究。在某些具有特定几何和功能结构的算术流形上，联合特征函数（Hecke- mass特征形式）表现出幂增长，这是物理模型既不能普遍预期也不能预测的。PI将寻找极限增长，并详细研究几种特定类型的算术流形的超范数和限制范数问题，以告知一般猜想和理解算术流形上的质量集中现象，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。在涉及字符和大水平自同构形式的数论问题中，深度和光滑方面与强导体或可分解导体有关，起着非常独特的作用。本文将使用非阿基米德分析、解析数论和谱理论研究强因子结构对l函数的非消失性、次凸性和矩的结构影响，以及涉及p-根解析波动的指数和。

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.