Residual Automorphic Representations and Automorphic L-functions

PI研究了剩余自同构表示的两个问题：第一，完备自同构L-函数的极点。第二，剩余自同构表示的局部分量的参数化。这是Arthur猜想的一部分，即用某个有限算术群的表示来表示离散谱。第一个问题是建立依附于一般尖点表示的所有完备自同构L函数的全纯，这些函数出现在Eisenstein级数的常数项中。PI使用朗兰兹-沙希迪方法，并观察到剩余自同构表示的局部分量是酉表示。这是可能的，因为由于在局部结果方面的最新进展，例如标准模猜想和沙希迪关于局部L-函数的全纯的猜想，我们得到了归一化局部缠绕算子的全纯和非零化。例如，作为与沙希迪的合作，PI建立了与GL_2的非单项尖点表示有关的第三次对称幂L函数的全纯，这一问题在过去20年中一直没有解决。利用逆定理，PI继续研究GL_2的尖顶表示的对称立方升力的存在性。第二个问题是对剩余自同构表示的局部分量进行参数化。这相当于将标准化的局部交织操作符的图像参数化。PI想要证明剩余自同构表示的局部分量来自环面的平凡特征，由对偶群和Springer对应的可区别的么等轨道来参数化。这一研究领域是数论的一部分，通常被称为朗兰兹程序。数论是研究整数性质的学科，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创造新数学的驱动力。朗兰兹计划是一种将数论与微积分联系起来的一般哲学；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刻性的，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.