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

高阶群上的自守形式：傅立叶系数、L 函数和算术

• 批准号: EP/T028343/1
• 负责人: Abhishek Saha
• 金额: $ 59.27万
• 依托单位: Queen Mary University of London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT028343%2F1
• 关键词: Automorphic; forms; higher; rank; groups

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.

拟议的研究在于接口的数论与代数，几何，分析和数学物理。在基本理论的启发下，我们提出了一种新的研究高阶群上自守形式的有力工具，以解决该领域中一些最深层次的开放问题.自守形式是李群上的高度对称函数，李群是现代数学中最重要的概念之一.它们是数论的关键，例如，理解整数系数的多项式方程，并且是该主题中许多最重要问题的中心。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（自守形式的一个例子）和椭圆曲线之间的深层联系。自守形式与它们的相关L-函数一起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结构的庞大网络--它是当今数学研究中最活跃的领域之一。此外，七个克莱“百万美元”千禧年奖问题中的两个都在自守L函数领域。自守形式也与数学物理的一些领域有联系，如量子混沌、弦理论和量子场论。在过去的世纪里，我们对模形式和马斯形式的详细理解有了相当大的进展，这是（秩1）群GL（2）上的两种自守形式。然而，在更高级别的案件中，进展有限得多。事实上，高阶群上自守形式的分析方面只是在最近几年才成为焦点，进展主要限于GL（3）等特殊情况。在更高级别的环境中，现有的方法和范式会崩溃，需要新的想法和创新的发展。该项目旨在围绕自守形式的傅立叶系数，周期公式，L函数和算术的思想圈取得深远的突破，以解决该领域中一些最重要和最实质性的问题。在一个显着偏离现有的工作在这一领域，我们将同时从分析，代数和算术方向处理这些问题。该项目在结果（新的“主定理”，使以前的几个结果下一个伞），方法（通过结合不同的方法框架）和领域（我们将汇集在一起的数学相对较少的相互作用的不同领域）的水平统一这些研究领域。最终，这项研究将为朗兰兹纲领与数论、几何、代数、分析和数学物理中的几个主题之间架起一座新的桥梁。此外，它将解决一些最实质性的问题，在该领域在更高的秩设置，如分布的广义傅立叶系数，量子唯一遍历性，超规范问题，次凸性，时刻的家庭的L-功能，和德利涅猜想的临界值的L-功能，以及开辟了许多途径，进一步探索。

项目成果

Superscars for arithmetic point scatters II

• DOI: 10.1017/fms.2023.33
• 发表时间: 2019-10
• 期刊: Forum of Mathematics, Sigma
• 作者: P. Kurlberg;S. Lester;Lior Rosenzweig

Weighted central limit theorems for central values of $L$-functions

$L$ 函数中心值的加权中心极限定理

• DOI: 10.4171/jems/1417
• 发表时间: 2024
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Bui H

ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2

2阶Siegel尖点形式的基本傅立叶系数

• DOI: 10.1017/s1474748021000542
• 发表时间: 2021
• 期刊: Journal of the Institute of Mathematics of Jussieu
• 影响因子: 0.9
• 作者: Jääsaari J

On the distribution of lattice points on hyperbolic circles

关于双曲圆上格点的分布

• DOI: 10.2140/ant.2021.15.2357
• 发表时间: 2021
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Chatzakos D