Automorphic differential equations and applications

自守微分方程及其应用

• 批准号: RGPIN-2021-04316
• 负责人: Sebbar, Abdellah
• 金额: $ 1.31万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741866
• 关键词: Automorphic; differential; equations; applications

My research program is in the area of number theory. This theory is linked to all mathematics, from the purest abstraction to fundamental applications in codes and cryptography. From the theoretical point of view, the last decades have seen spectacular advances around the work of Wiles which led to the solution of Fermat's theorem, as well as advances in analytic number theory. There are also applications in theoretical physics and in cryptographic schemes against quantum computers. Number theory is also a powerful tool in developing computing methods. In this proposal I study a kind of differential equations that are closely related to automorphic forms and are called automorphic differential equations. In nature, they are types of Schrodinger equations with an automorphic potential in the same way that the Lamé equations are Schrodinger equations with an elliptic potential. The automorphic potential is in fact an automorphic form of weight four. Since the space of these forms for the modular group is one-dimensional, our equations depend on a complex parameter. For each class of these parameters, the solutions are completely different in the sense that they possess different kinds of symmetry. However, they all share one common property: They are all equivariant with respect to a representation of the modular group. Thus the classification of these representations yields various types of solutions to the automorphic differential equations. Various tools from complex analysis, algebraic geometry and representation theory are used. This has many applications, the most relevant of which to this proposal is about making progress toward solving a conjecture that states that the zeros of the weight two Eisenstein series are transcendental numbers. This is an important conjecture which turns out to be a particular case of two famous conjectures by Grothendieck/André and by Nesterenko/Bertrand. This research program will also provide high-quality training to HQP allowing them to develop the skills that will make them competitive candidates for academic and non-academic jobs.

我的研究项目是在数论领域。这个理论与所有的数学都有联系，从最纯粹的抽象到代码和密码学的基本应用。从理论的角度来看，在过去的几十年里，怀尔斯的工作取得了惊人的进展，他的工作导致了费马定理的求解，分析数论也取得了进展。在理论物理和针对量子计算机的加密方案中也有应用。数论也是开发计算方法的有力工具。在这个提议中，我研究了一类与自同构形式密切相关的微分方程，称为自同构微分方程。在本质上，它们是具有自同构势的薛定谔方程，就像lam<s:1>方程是具有椭圆势的薛定谔方程一样。自同构势实际上是权值4的自同构形式。由于模群的这些形式的空间是一维的，我们的方程依赖于一个复参数。对于这些参数的每一类，解都是完全不同的，因为它们具有不同的对称性。然而，它们都有一个共同的性质：它们对于模群的表示都是等变的。因此，这些表示的分类产生了自同构微分方程的各种类型的解。从复杂分析，代数几何和表示理论的各种工具被使用。这有很多应用，与这个提议最相关的是在解决一个猜想方面取得进展，这个猜想表明权2爱森斯坦级数的零是超越数。这是一个重要的猜想，它被证明是格罗滕迪克/安德烈和涅斯捷连科/伯特兰两个著名猜想的一个特例。该研究项目还将为HQP提供高质量的培训，使他们能够发展技能，使他们在学术和非学术工作中具有竞争力。