Topics in the Theory of Automorphic Representations

研究者研究了现代自同构形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型的内蕴对称结构。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性来刻画。近年来，周期成为刻画自同构型和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同构形式族，特别是退化的尖顶自同构形式。为了理解Ramanujan猜想在估计自同构形的傅里叶系数上的推广，I.Piatetski-Shapiro在1980年代初系统地研究了这种退化的尖顶自同构形的第一个例子。这些以相关周期为特征的新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工作中提出的基本结构相同。第二个问题是研究当地和全球的朗兰兹互惠法律。这位研究人员和他的同事打算通过研究局部伽马函数和利用局部和全局自同构形式理论的最新进展来建立经典群的局部朗兰兹互易定律。研究人员和他的同事研究自同构形式理论，这是一种抽象的先验理论，描述了声音和波等重复现象。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，这位研究人员和他的同事可以从几何、分析和数论方面研究对象之间的相互关系。这一理论在A.Wiles最近对费马大定理的求解中起到了至关重要的作用，也对编码理论和密码学的最新发展起到了重要作用。研究人员和他的合作者在该项目中研究了自同构形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.