Automorphic L-Functions, Endoscopy, and Representation Theory

沙希迪教授将继续研究二维向量空间上一般线性群上非单项尖点形式的对称立方L函数的全纯。这是建立四维空间上一般线性群的对称立方提升存在性的第一步。调查者还将考虑所有L函数的极点问题--这些函数是由先前与R.朗兰兹共同建立的方法得到的。也在考虑这些函数的有限条上的序的有界性或有限性。在一个相关的项目中，沙希迪教授将试图通过建立归一化交织算子所满足的新恒等式，将亚瑟迹公式的应用增加到最一般的情况。这个项目涉及数论和表象理论中的问题。从历史上看，数学中有很多例子，其中关于数的基本性质的重要信息被编码在微积分的函数中。在过去的二十年里，数论的一个重要进展是人们越来越多地理解这些例子是一个非常普遍的理论的一部分。完整的一般理论仍然是一个相当大的数学谜团。然而，它是如此强大，以至于每向一般理论迈出一步，其影响就会在数论和表示论的各个部分感受到。L函数是编码数学家所需要的信息的函数，这些信息出现在称为函数极点的点上。在这个项目中，沙希迪教授将研究从函数中提取这些信息的几种方法。

Professor Shahidi will continue his study of the holomorphy of symmetric cube L-functions for non-monomial cusp forms on the general linear group on dimension two vector spaces. This is the first step in establishing the existence of the symmetric cube lift to the general linear group on dimension four spaces. The investigator will also consider the questions of poles of all L-functions that are obtained from a method previously established in conjunction with R. Langlands. Also under consideration is the boundedness or finiteness of order on finite strips of theses functions. In a related project Professor Shahidi will attempt to increase the applications for Arthur's trace formula to the most general cases by establishing new identities satisfied by normalized intertwining operators. This project involves questions in number theory and representation theory. Historically there have been many examples in mathematics where important information about the fundamental properties of numbers is encoded in the functions of calculus. An important advance in number theory over the past twenty years has been the increased understanding that these examples were part of a very general theory. The full general theory is still quite a mathematical mystery. It is so powerful, however, that each time a step toward the general theory is made, the repercussions are felt in all parts of number theory and representation theory. L-functions are the functions that often encode the information that number theorists need, and the information appears at points called the poles of the function. In this project, Professor Shahidi will investigate several methods of extracting this information from the function.