Residual Automorphic Representations and Automorphic L-functions

PI研究了残差自同构表示的两个问题：一是完全自同构l函数的极点问题。其次，残差自同构表示局部分量的参数化。这是Arthur关于用一定无限算术群的表示来参数化离散谱的猜想的一部分。第一个问题是建立在爱森斯坦级数常项中所有附于一般倒形表示的完备自同构l函数的全纯性。该PI采用Langlands-Shahidi方法，并观察残差自同构表示的局部分量是酉表示。这是可能的，因为最近在局部结果上的进展，如关于局部l函数全纯的标准模猜想和Shahidi猜想，使人们有了归一化局部交织算子的全纯性和不灭性。例如，PI与Shahidi共同建立了GL_2的非单项式倒形表示上的第三对称幂l函数的全纯性，这个问题在过去20年里一直没有得到解决。PI继续利用逆定理研究GL_2的倒丘表示的对称立方升的存在性。第二个问题是对残差自同构表示的局部分量进行参数化。这相当于参数化归一化局部缠结算子的图像。PI想要证明残差自同构表示的局部分量来自环面的平凡特征，是由对偶群和施普林格对应的不同的单幂轨道参数化的。这个研究领域是数论的一部分，通常被称为朗兰兹程序。数论是对整数性质的研究，是数学中最古老的分支。从一开始，数论中的问题就成为在该学科的其他不同部分中创造新数学的推动力。朗兰兹纲领是一种将数论与微积分联系起来的普遍哲学；它体现了研究整数的现代方法。现代数论是非常技术性和深奥的，但它在理论计算机科学和编码理论等领域有着惊人的应用。

The PI studies two problems which come from the study of residual automorphic representations: First, poles of completed automorphic L-functions. Second, parametrization of the local components of residual automorphic representations. It is a part of Arthur's conjecture on parametrizing the discrete spectrum in terms of representations of a certain profinite arithmetic group. The first problem is to establish the holomorphy of all completed automorphic L-functions attached to generic cuspidal representations which appear in the constant terms of Eisenstein series. The PI uses Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations. This is possible because one has the holomorphy and non-vanishing of normalized local intertwining operators due to recent progress on local results, such as standard module conjecture and Shahidi's conjecture on the holomorphy of local L-functions. For example, the PI established, as a joint work with Shahidi, the holomorphy of the third symmetric power L-functions attached to non-monomial cuspidal representations of GL_2, which had been unsolved last 20 years. The PI continues to work on the existence of the symmetric cube lift of cuspidal representations of GL_2, using the converse theorem. The second problem is to parametrize the local components of residual automorphic representations. This amounts to parametrizing the image of normalized local intertwining operators. The PI wants to show that the local components of the residual automorphic representations coming from the trivial character of the torus, are parametrized by the distinguished unipotent orbits of the dual group and Springer correspondence.This research area is in a part of number theory generally known as the Langlands program. Number theory is the study of the properties of whole numbers and is the oldest branch of mathematics. From the beginning, problems in number theory have served as a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.