Automorphic forms: arithmetic and analytic interfaces

这个项目的目标是更好地理解高阶自守形式的性质，特别是在涉及高分支的情况下。自守形式是朗兰兹纲领的核心对象，朗兰兹纲领是一个庞大的定理和猜想网络，将来自数论、表示论和几何的概念联系起来。自守形式的最简单的例子包括狄利克雷特征标和经典模形式，这两者在现代数学中都被证明是非常重要的。更一般地，自守形式是复值函数，可以自然地被视为称为自守表示的表示中的向量。这种观点允许人们将自守形式与任何还原代数群联系起来。从不同的观点来看，自守形式包括（作为特殊情况）算术流形上拉普拉斯算子的本征函数。这种观点允许人们从分析、光谱理论和量子力学中引入一系列额外的观点。自守形式和与之相关的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代数整数的三元二次形式表示的工作。现代数论的一个中心主题是理解自守形式及其相关L函数在一个或多个定义参数变化时的关键性质。这些参数的有限或非阿基米德部分可以通过一个基本的算术量来捕获，称为导体或水平（以下用N表示），用于测量其总分支（或有限素数的复杂性）。层次出现在附加L函数的泛函方程中，同时（本质上）描述了自守形式所处的算术流形，与阿基米德方面相比，关于自守形式的解析问题的层次方面的研究进展相对较少，特别是在高阶方面。在这个项目中，学生将调查与上述主题相关的关键问题。所使用的工具将是一个混合的代数以及解析数论，连同代表性理论的padic集团。要解决的具体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）。

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.)