Automorphic forms on higher rank groups: Fourier coefficients, L-functions, and arithmetic

所提出的研究是数论与代数、几何、分析和数学物理的结合。在基本猜想的激励下，我们建议开发强大的新工具来研究高阶群上的自同构形式，以接近该领域中一些最深的开放问题。自同构形式是李群上的高度对称函数，是现代数学中最重要的概念之一。它们是数论的关键，例如，理解具有整数系数的多项式方程，并且是该学科中许多最重要问题的核心。例如，安德鲁·怀尔斯爵士在1995年对费马大定理的证明依赖于模形式（自同构形式的一个例子）和椭圆曲线之间的深层联系。与它们相关的l函数一起，自同构形式也是朗兰兹纲领的中心对象。朗兰兹纲领是一个连接代数、几何、数论和分析的定理和猜想的庞大网络，是当今数学研究中最活跃的领域之一。此外，七个Clay“百万美元”千禧年奖问题中的两个是在自同构l函数领域。自同构形式也与数学物理的几个领域有联系，比如量子混沌、弦理论和量子场论。在过去的一个世纪里，我们对（秩1）群GL(2)上的两种自同构形式——模形式和质量形式的详细认识有了相当大的进展。然而，在级别较高的案件中取得的进展要有限得多。事实上，高阶群上自同构形式的分析方面直到最近几年才成为人们关注的焦点，其进展主要局限于GL(3)等特殊情况。在更高级别的环境中，现有的方法和范式被打破，需要发展新的想法和创新。这个项目的目标是在自同构形式的傅里叶系数、周期公式、l函数和算法等方面取得意义深远的突破，以解决该领域一些最重要和最实质性的问题。我们将从解析、代数和算术方向同时处理这些问题，这与该领域的现有工作有很大的不同。这个项目将这些研究领域在结果（新的“主定理”，将以前的几个结果整合在一起）、方法（通过结合不同的方法框架）和领域（我们将把相互作用相对较少的不同数学领域整合在一起）的层面上统一起来。这将使我们能够取得以前无法取得的进展。最终，这项研究将在朗兰兹计划和数论、几何、代数、分析和数学物理等几个主题之间架起一座新的桥梁。此外，它将解决该领域在更高等级设置中的一些最重要的问题，如广义傅里叶系数的分布、量子唯一遍历性、超范数问题、次凸性、l函数族的矩和Deligne关于l函数临界值的猜想，并为进一步探索开辟了许多途径。

The proposed research lies at the interface of number theory with algebra, geometry, analysis and mathematical physics. Motivated by fundamental conjectures, we propose to develop powerful new tools to investigate automorphic forms on higher rank groups in order to approach some of the deepest open problems in the field.Automorphic forms are highly symmetric functions on Lie groups that constitute one of the most important concepts in modern mathematics. They are key to number theory, e.g., understanding polynomial equations with integer coefficients, and lie at the centre of many of the most important problems in the subject. For instance, Sir Andrew Wiles' proof of Fermat's Last Theorem in 1995 relied on a deep connection between modular forms (an example of automorphic forms) and elliptic curves. Together with their associated L-functions, automorphic forms are also central objects in the Langlands programme - a vast web of theorems and conjectures connecting algebra, geometry, number theory, and analysis - which is one of the most active areas of mathematical research today. Additionally, two of the seven Clay "million dollar" Millennium Prize Problems lie in the area of automorphic L-functions. Automorphic forms also have connections to several areas of mathematical physics, such as quantum chaos, string theory, and quantum field theory.Over the last century, there has been considerable progress in our detailed understanding of modular forms and Maass forms, which are the two types of automorphic forms on the (rank 1) group GL(2). However, progress in the higher rank cases has been much more limited. Indeed, the analytic aspects of automorphic forms on higher-rank groups has come into focus only in the last few years, with progress largely limited to special cases such as GL(3). In higher rank settings, existing methods and paradigms break down requiring the development of new ideas and innovations. This project sets out to make far-reaching breakthroughs relating to the circle of ideas around Fourier coefficients of automorphic forms, period formulas, L-functions, and arithmetic to resolve some of the most important and substantial problems in the field. In a significant departure from existing work in this field, we will approach these problems simultaneously from the analytic, algebraic, and arithmetic directions. This project unifies these research areas at the level of results (new "master theorems" that bring several previous results under one umbrella), methods (by combining distinct methodological frameworks), and fields (we will bring together different fields of mathematics which have seen relatively little interaction). This will allow us to make advances that were previously inaccessible.Ultimately, this research will provide a new bridge between the Langlands programme and several topics in number theory, geometry, algebra, analysis, and mathematical physics. Moreover, it will resolve some of the most substantial problems in the field in higher rank settings such as the distribution of generalised Fourier coefficients, Quantum Unique Ergodicity, the sup-norm problem, subconvexity, moments of families of L-functions, and Deligne's conjecture on critical values of L-functions, as well as open up numerous avenues for further exploration.