New bounds towards Fourier coefficients of Siegel modular forms

西格尔模形式傅里叶系数的新界限

• 批准号: EP/W001160/1
• 负责人: Abhishek Saha
• 金额: $ 10.27万
• 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: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW001160%2F1
• 关键词: New; bounds; towards; Fourier; coefficients

Automorphic forms are highly symmetric functions that constitute one of the most important concepts in modern mathematics. 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. A key way in which automorphic forms can be understood is via their Fourier coefficients. Basic questions about Fourier coefficients of automorphic forms can contain an incredible amount of deep mathematics and can be extremely hard. For example, Ramanujan's conjecture (made in 1916) regarding an upper bound for the size of Fourier coefficients of modular forms was finally proved by Deligne in 1974, as a consequence of his deep, Fields medal winning work in arithmetic geometry. A very natural generalization of the (classical) modular forms is given by the Siegel modular forms, which were first investigated by Carl Ludwig Siegel in the 1930s. They are of great importance in number theory and the Langlands programme, and also have applications to physics and information technology. To give an example, Wiles' proof of Fermat's last theorem relies on a deep connection between modular forms and elliptic curves; the generalization of this to one dimension up (the so-called paramodular conjecture, which is a hot topic currently) involves Siegel modular forms.The main goal of this project is to prove new bounds towards the Fourier coefficients of (cuspidal) Siegel modular forms and thus make progress towards the famous Resnikoff-Saldana conjecture, a problem that has been open for almost 50 years. The successful completion of this project will lead to new improved understanding of Siegel modular forms, and it will demonstrate for the first time deep links between the Resnikoff-Saldana conjecture and other central conjectures in number theory. This will open up many avenues of further exploration.

自动形式是高度对称的功能，构成了现代数学中最重要的概念之一。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。可以通过其傅立叶系数理解自动形式的关键方法。关于汽车形式的傅立叶系数的基本问题可能包含令人难以置信的深度数学，并且非常困难。例如，Ramanujan的猜想（于1916年制作）关于模块化形式的傅立叶系数大小的上限，最终在1974年证明了他的深刻田野奖牌获奖作品。 Siegel模块化形式给出了（经典）模块化形式的非常自然的概括，这是Carl Ludwig Siegel在1930年代首次研究的。它们在数字理论和兰兰兹计划中非常重要，并且还适用于物理和信息技术。举一个例子，威尔斯证明了费玛特的最后定理依赖于模块化形式和椭圆曲线之间的深厚联系。这对一个维度的概括（当前是一个热门话题）涉及Siegel模块化形式。该项目的主要目的是为（Cuspidal）Siegel模块化形式的傅立叶系数证明新的界限，从而使著名的Resnikoff-Saldana Issuption进步，以至于曾经是一个问题，因为它已经为几乎开放了50年的问题，几乎是50岁。该项目的成功完成将导致对Siegel模块化形式的新理解，并将首次证明Resnikoff-Saldana猜想与数量理论中其他中心猜想之间的深入联系。这将开辟许多进一步探索的途径。

项目成果

On Fourier Coefficients and Hecke Eigenvalues of Siegel Cusp Forms of Degree 2

• DOI: 10.1093/imrn/rnac316
• 发表时间: 2022-07
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Biplab Paul;A. Saha