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.

自守形式是构成现代数学中最重要概念之一的高度对称函数。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。自守形式可以理解的一个关键方法是通过它们的傅立叶系数。关于自守形式的傅立叶系数的基本问题可能包含令人难以置信的大量深度数学，并且可能非常困难。例如，拉马努金的猜想（在1916年）关于上限的大小傅立叶系数的模块化形式终于证明了德利涅在1974年，由于他的深，菲尔兹奖获奖工作算术几何。西格尔模形式是（经典）模形式的一个非常自然的推广，它首先由卡尔·路德维希·西格尔在20世纪30年代研究。它们在数论和朗兰兹纲领中具有重要意义，在物理学和信息技术中也有应用。举个例子，怀尔斯对费马大定理的证明依赖于模形式和椭圆曲线之间的深刻联系;将其推广到一维（所谓的paramodular猜想，这是一个热门话题目前）涉及西格尔模形式。这个项目的主要目标是证明新的界限对傅立叶系数的（尖）西格尔模块化的形式，从而取得进展，对著名的Resnikoff-Saldana猜想，一个问题，已开放了近50年。这个项目的成功完成将导致对西格尔模形式的新的理解，并将首次展示Resnikoff-Saldana猜想与数论中其他中心理论之间的深层联系。这将为进一步探索开辟许多途径。

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