Topics in the Theory of Automorphic Representations

调查研究两个基本问题，在现代理论的自守形式。 第一个问题是关于数域上约化群的自守形式的内在对称结构。 这种自守形式的结构可以通过存在各种自守形式的周期积分来表征。 周期最近成为刻画自守形式和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，研究者和他的同事们打算用周期来刻画新的自守型族，特别是退化尖点自守型族。 第一个例子，这种退化尖点自守形式系统地研究了由I。Piatetski-Shapiro在1980年代早期为了理解Ramanujan猜想在自守形式的Fourier系数估计上的推广而提出的。 这些新的自守形式族具有相应的周期特征，它们的基本结构与J.亚瑟关于约化群的平方可积自守形式空间上的一般迹公式的工作所建议的自守形式族相同. 第二个问题是研究局域和全局朗兰兹互易律。 研究者和他的同事们打算通过研究局部伽马函数并利用自守形式的局部和整体理论的最新进展来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守形式理论，这是一种抽象的先验理论，描述了重复的现象，如声音和波。 作为现代数学的基石，它为数学的相关领域提供了基础，并应用于计算机网络理论和物理学中的弦理论。 利用这一理论，研究者和他的同事可以从几何、分析和数论的角度来考察物体之间的相互关系。 这一理论在最近A.怀尔斯，也一直是重要的编码理论和密码学的最新发展。 研究者和他的合作者在这个项目中研究自守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.