Automorphic L-Functions, Endoscopy, and Representation Theory

Shahidi教授将继续研究二维向量空间上一般线性群上非单项式尖形的对称立方l函数的全纯性。这是建立四维空间上一般线性群的对称立方提升的存在性的第一步。研究者还将考虑所有l函数的极点问题，这些问题是由先前与R. Langlands联合建立的方法获得的。也要考虑这些函数在有限条上的有界性或序的有限性。在一个相关的项目中，Shahidi教授将尝试通过建立归一化交织算子所满足的新恒等式，将Arthur的迹公式的应用范围扩大到最一般的情况。这个项目涉及数论和表示理论的问题。历史上，在数学中有很多例子，关于数字基本属性的重要信息被编码在微积分的函数中。在过去的二十年里，数论的一个重要进步是，人们越来越认识到这些例子是一个非常普遍的理论的一部分。完整的广义理论在数学上仍然是个谜。然而，它是如此强大，以至于每当向一般理论迈出一步，就会在数论和表示理论的各个部分感受到反响。l函数是通常编码数论学家需要的信息的函数，这些信息出现在称为函数极点的点上。在这个项目中，Shahidi教授将研究几种从函数中提取这些信息的方法。

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.