Automorphic forms: arithmetic and analytic interfaces

自守形式：算术和分析接口

• 批准号: 2612135
• 金额: --
• 依托单位: Queen Mary University of London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2612135
• 关键词: Automorphic; forms; arithmetic; analytic; interfaces

The goal of this project is to better understand the properties of automorphic forms of higher rank, especially in situations involving high ramification. Automorphic forms are central objects in the Langlands program, a vast web of theorems and conjectures that connects concepts coming from number theory, representation theory and geometry. The simplest examples of automorphic forms include Dirichlet characters and classical modular forms, both of which have proved to be of profound importance in modern mathematics. More generally, automorphic forms are complex valued functions that can be naturally viewed as vectors inside representations known as automorphic representations. This viewpoint allows one to associate automorphic forms to any reductive algebraic group. From a different point of view, automorphic forms include (as special cases) eigenfunctions of Laplacians on arithmetic manifolds. This viewpoint allows one to bring in a whole range of additional perspectives coming from analysis, spectral theory and quantum mechanics. Automorphic forms and the L functions attached to them have been key ingredients in the solutions of many famous and difficult problems, such as Wiles' proof of Fermat's last Theorem and Duke's work on the representations of algebraic integers by ternary quadratic forms.A central theme in modern number theory is to understand key properties of automorphic forms and their associated L functions as one or more of their defining parameters vary. The finite or non archimedean part of these parameters can be captured by a fundamental arithmetic quantity called the conductor or level (henceforth denoted by N) that measures its total ramification (or complexity at finite primes). The level appears in the functionalequation of the attached L function, as well as (essentially) describes the arithmetic manifold that the automorphic form lives on. Compared to the archimedean aspect, there has been relatively little progress in the level aspect versions of analytic problems about automorphic forms, especially in higher rank. In this project, the student will investigate key questions related to the themes described above. The tools used will be a mix of algebraic as well as analytic number theory, together with representation theory of p adic groups. The specific problems to be solved will depend on the interests of the student (possible examples include sup norms and other L^p norms,period formulas, etc.)

该项目的目标是更好地理解更高阶自同构形式的属性，特别是在涉及高分支的情况下。自守形式是朗兰兹纲领中的核心对象，朗兰兹纲领是一个庞大的定理和猜想网络，连接着来自数论、表示论和几何的概念。自守形式最简单的例子包括狄利克雷特征和经典模形式，这两者都已被证明在现代数学中具有深远的重要性。更一般地说，自同构形式是复值函数，可以自然地将其视为称为自同构表示的表示内部的向量。这一观点允许人们将自同构形式与任何还原代数群联系起来。从不同的角度来看，自同构形式包括（作为特殊情况）算术流形上拉普拉斯算子的本征函数。这种观点允许人们引入来自分析、光谱理论和量子力学的一系列额外观点。自守形式和附加的 L 函数是解决许多著名和困难问题的关键要素，例如怀尔斯对费马大定理的证明和杜克关于三元二次形式的代数整数表示的工作。现代数论的目的是理解自同构形式的关键属性及其相关的 L 函数，因为它们的一个或多个定义参数发生变化。这些参数的有限或非阿基米德部分可以通过称为导体或能级（以下用 N 表示）的基本算术量来捕获，该算术量测量其总分支（或有限素数处的复杂性）。该级别出现在附加 L 函数的函数方程中，并且（本质上）描述了自守形式所依赖的算术流形。与阿基米德方面相比，有关自同构形式的分析问题的水平方面版本进展相对较小，尤其是在更高阶方面。在这个项目中，学生将研究与上述主题相关的关键问题。使用的工具将是代数和解析数论以及 padic 群表示论的混合。具体要解决的问题将取决于学生的兴趣（可能的例子包括sup范数和其他L^p范数、周期公式等）