New bounds towards Fourier coefficients of Siegel modular forms

自同构形式是高度对称的函数，是现代数学中最重要的概念之一。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。理解自同构形式的一个关键方法是通过它们的傅里叶系数。关于自同构形式的傅里叶系数的基本问题可能包含令人难以置信的深度数学，并且可能非常困难。例如，拉马努金关于模形式的傅里叶系数大小的上界的猜想（1916年提出）最终在1974年被德列涅证明，这是他在算术几何领域获得菲尔兹奖的深刻工作的结果。西格尔模形式给出了（经典）模形式的一个非常自然的推广，它是由卡尔·路德维希·西格尔在20世纪30年代首次研究的。它们在数论和朗兰兹程序中非常重要，在物理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层联系；将其推广到一维以上（所谓的旁模猜想，这是当前的一个热门话题）涉及到西格尔模形式。该项目的主要目标是证明（cusidal） Siegel模形式的傅里叶系数的新界限，从而在著名的Resnikoff-Saldana猜想方面取得进展，这个问题已经开放了近50年。这个项目的成功完成将导致对西格尔模形式的新的改进理解，它将首次证明Resnikoff-Saldana猜想与数论中其他中心猜想之间的深刻联系。这将开辟许多进一步探索的途径。

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.