Automorphic differential equations and applications

我的研究项目是数论。这一理论与所有数学有关，从最纯粹的抽象到代码和密码学中的基本应用。从理论的角度来看，在过去的几十年里，威尔斯的工作取得了惊人的进步，导致了费马定理的解决，以及解析数论的进步。在理论物理和针对量子计算机的密码方案中也有应用。数论也是发展计算方法的有力工具。在这个方案中，我研究了一类与自同构形式密切相关的微分方程，我们称之为自同构方程。本质上，它们是具有自同构势的薛定谔方程，就像LAMé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重4的自同构形。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，它们的解完全不同，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：它们关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常微分方程解的各种类型。运用了复变分析、代数几何和表象理论等多种工具。这有许多应用，其中与这一提议最相关的是在解决一个猜想方面取得进展，该猜想声称权重为2的艾森斯坦级数的零点是超越数。这是一个重要的猜想，是Grothendieck/André和Nester enko/Bertrand提出的两个著名猜想的特例。该研究计划还将为HQP提供高质量的培训，使他们能够发展技能，使他们成为学术和非学术工作的有竞争力的候选人。

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.