Topics in the Theory of Automorphic Representations

研究者研究了现代自同构形式理论中的两个基本问题。第一个问题是关于数域上约化群的自同构形式的本征对称结构。这种自同构形式的结构可以用自同构形式的各种周期积分的存在来表征。周期近来成为刻画自同构形式和l -函数的各种基本性质的一个非常有用的工具。在这个项目中，研究者和他的同事打算用周期来表征新的自同构形式，特别是退化的逆自同构形式。为了理解Ramanujan猜想关于自同构形式傅里叶系数估计的推广，I. Piatetski-Shapiro在20世纪80年代初系统地研究了这类退化倒角自同构形式的第一个例子。这些新的具有相关周期特征的自同构形式族有望与J. Arthur在约化群的平方可积自同构形式空间上的一般迹公式中提出的自同构形式族具有相同的基本结构。第二个问题是研究局部和全局朗兰兹互易律。研究者和他的同事打算通过研究局部伽马函数，并利用自同构形式的局部和全局理论的最新进展，建立经典群的局部朗兰兹互易律。研究者和他的同事们研究自同构形式理论，这是一种抽象的先验理论，描述声音和波等重复现象。它是现代数学的基石，为数学的相关领域提供了基础，并应用于物理中的计算机网络理论和弦理论。利用这一理论，研究者和他的同事可以从几何、分析和数论中检查对象之间的互反关系。该理论在怀尔斯最近对费马大定理的求解中发挥了重要作用，对最近的编码理论和密码学的发展也起了重要作用。研究者和他的合作者在项目中研究了自同构形式的内在对称结构和超越不变量。

The investigator studies two fundamental problems in the modern theory of automorphic forms. The first problem is concerned with the intrinsic symmetric structures of automorphic forms of reductive groups over number fields. Such structures of automorphic forms can be characterized by the existence of various period integrals of automorphic forms. Periods become recently a very useful tool to characterize various basic properties of automorphic forms and L-functions. In this project, the investigator and his colleagues intend to use periods to characterize new families of automorphic forms, especially, the degenerate cuspidal automorphic forms. The first example of such degenerate cuspidal automorphic forms is systematically studied by I. Piatetski-Shapiro in early 1980's in order to understand the generalization of the Ramanujan conjecture on estimate of Fourier coefficients of automorphic forms. These new families of automorphic forms characterized by the relevant periods are expected to share the basic structures with the ones suggested by J. Arthur from his work on general trace formula over the space of square integrable automorphic forms of reductive groups. The second problem is to study local and global Langlands reciprocity laws. The investigator and his colleague intend to establish local Langlands reciprocity law for classical groups by studying local gamma functions and using recent progress in the local and global theory of automorphic forms.The investigator and his colleagues study the theory of automorphic forms, an abstract, transcendental theory that describes repeating phenomena like sound and waves. A cornerstone of modern mathematics, it provides the foundation for relevant areas of mathematics and has applications to computer network theory and string theory in physics. Using this theory, the investigator and his colleagues can examine the reciprocal relations among objects from geometry, analysis, and number theory. The theory played an essential role in recent solution of the Fermat's last theorem by A. Wiles, and has also been important to the recent development in coding theory and cryptology. The investigator and his collaborators study in the project the intrinsic symmetric structures and transcendental invariants of automorphic forms.