Automorphic forms: arithmetic and analytic interfaces

这个项目的目标是更好地理解高阶自同构形式的性质，特别是在涉及高分支的情况下。自同构形是朗兰兹计划的中心对象，朗兰兹计划是一个庞大的定理和猜想网络，将数论、表示论和几何中的概念联系在一起。最简单的自同构形式的例子包括狄里克莱特特征标和经典的模形式，这两种形式都已被证明在现代数学中具有深远的重要性。更广泛地说，自同构形式是复值函数，可以自然地被视为称为自同构表示的表示内的矢量。这一观点允许人们将自同构形式与任何约化代数群联系起来。从不同的角度来看，自同构形式包括(作为特例)算术流形上的拉普拉斯的特征函数。这一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附加观点。自同构形及其附带的L函数是解决许多著名和困难问题的关键因素，如Wiles对费马大定理的证明和Duke关于代数整数用三元二次型表示的工作。现代数论的一个中心主题是理解自同构形及其关联的L函数的关键性质，因为它们的一个或多个定义参数不同。这些参数的有限或非阿基米德部分可以通过称为导体或电平(此后用N表示)的基本算术量来捕捉，该基本算术量测量其全部分支(或在有限素数处的复杂性)。水平出现在附加的L函数的函数方程中，并且(本质上)描述了自同构形赖以生存的算术流形。与阿基米德方面相比，关于自同构形式的分析问题的水平方面版本的进展相对较小，特别是在更高的等级上。在这个项目中，学生将调查与上述主题相关的关键问题。所使用的工具将是代数和解析数论的混合，以及p进群的表示理论。需要解决的具体问题将取决于学生的兴趣(可能的例子包括超级规范和其他L^p规范、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.)