Problems in the Spectral Theory of Automorphic Forms and Number Theory

本研究的目的是研究某些l -函数的临界线上，并研究某些完全可积系统上拉普拉斯算子谱的精细结构。l函数是复解析函数，在复平面的定域上由幂级数给出，但它们最有趣的行为是在它们的临界线上，这条线属于解析延拓的区域。研究由有限面积双曲曲面的尖形产生的Rankin-Selberg型l函数，涉及到一般的Ramanujan猜想。所使用的方法属于自同构形式的谱理论。光谱精细结构中的问题最近引起了从事量子混沌研究的物理学家的注意。中心问题是特征值之间间隔的分布。希望将解析数论的工具引入到完全可积系统的对相关函数的研究中。本研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。

Stopple The aim of this investigation is to study certain L-functions on their critical line and to study the finer structure of the spectrum of the Laplace operator on certain completely integrable systems. The L-functions are complex analytic functions, given by power series in a domain of the complex plane, but their most interesting behavior is on their critical line, which belongs to the region of analytic continuation. The study of L-functions of Rankin-Selberg type that arise from cusp forms of finite area hyperbolic surfaces is related to the general Ramanujan conjecture. The methods to be used belong to the spectral theory of automorphic forms. The problems in the finer structure of the spectrum have recently attracted attention among physicists working on Quantum Chaos. The central problem is the distribution of the spacings between the eigenvalues. The hope is to bring tools from analytic number theory to the study of the pair correlation function of completely integrable systems. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.