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.

拟议的研究在于数字理论与代数，几何，分析和数学物理学的界面。由基本猜想的激励，我们建议开发强大的新工具，以调查更高排名组的自身形式，以解决该领域中一些最深层的开放问题。自动形态形式是构成现代数学中最重要概念之一的谎言组的高度对称功能。它们是数字理论的关键，例如，使用整数系数理解多项式方程，并位于该主题上许多最重要问题的中心。例如，安德鲁·威尔斯爵士（Andrew Wiles）在1995年对费马特（Fermat）的最后一个定理的证明依赖于模块化形式（自动形式的一个例子）和椭圆曲线之间的深厚联系。与其相关的L功能一起，自动形式也是Langlands计划中的中心对象 - 连接代数，几何，数字理论和分析的庞大的定理和猜想 - 这是当今数学研究中最活跃的领域之一。此外，七个粘土“百万美元”千年奖项问题中有两个在于自动型L功能领域。自动形式也与数学物理学的几个领域有联系，例如量子混乱，弦理论和量子场理论。在上个世纪，我们对模块化形式和MAASS形式的详细理解已经取得了很大的进步，这是（等级1）GL（2）的两种类型的自动形式。但是，在较高排名的情况下的进展要受到更大的限制。实际上，自动形式对高级群体的分析方面仅在过去几年才开始焦点，进展很大程度上仅限于特殊情况（例如GL（3））。在较高的等级设置中，现有的方法和范式分解了需要开发新思想和创新的方法。该项目旨在实现与自动形式，周期公式，l功能和算术的傅立叶系数有关的思想循环的深远突破，以解决该领域中一些最重要和最重要的问题。在与该领域的现有工作的显着背道而驰，我们将从分析，代数和算术方向同时解决这些问题。该项目将这些研究领域统一的结果级别（新的“主定理”将以前的几个结果带入一个保护伞），方法（通过结合不同的方法框架）和领域（我们将汇总到相对较少的相互作用的不同数学领域）。这将使我们能够取得以前无法访问的进步。最终，这项研究将在兰兰兹计划与数字理论，几何，代数，分析和数学物理学的几个主题之间提供新的桥梁。此外，它将在较高排名的设置中解决该领域中一些最重要的问题，例如广义傅立叶系数的分布，量子独特的奇特性，SUP-NORM问题，子概念，L-Intunctions的时刻，DeLigne的瞬间以及deligne的猜想对L功能的关键价值，以及众多探索的探索。

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 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

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