L-functions of automorphic forms and their variants

经典自守形式是处理困难数论问题的有力工具。它们提供了数学问题研究的分析，代数和几何方面之间的联系，因此，它们是数论中主要研究方案的核心，例如朗兰兹方案。对于这些联系至关重要的是与自守形式相关的某些函数，称为L-函数，这是一些最重要的数学结构的主题。近年来，对自守形式理论的研究导致了自守形式及其L-函数的变体的研究，例如拟模形式，调和Maass形式，mock模形式，高阶模形式和多重Dirichlet级数。在大多数情况下，引入这些对象的动机不仅仅是为了推广经典的自守形式及其L函数，而是为了获得新的工具来解决已经提出的数论问题。与这些新对象相关的技术反过来又提出了新的有趣的问题，并突出了原始激励问题之外的联系。例如，调和Maass形式和模形式的理论已被用于解决数的划分问题，高阶模形式已被应用于物理学中的渗透理论问题。由于这些技术最近才被发现，它们导致了许多非常有趣的开放问题，例如如何构建模拟模形式编码特定的划分函数，如何确定高阶形式的算术性质，或者如何使用多重狄利克雷级数理论来约束黎曼zeta函数的矩。这种类型的问题，对于古典自守形式中的突出问题和新学科本身的进一步发展都是高度相关的。因此，这些问题中的许多都非常适合博士项目。

Classical automorphic forms are a powerful tool for handling difficult number theoretic problems. They provide links between analytic, algebraic and geometric aspects of the study of arithmetic problems and, as such, they are at the heart of the major research programmes in Number Theory, e.g. Langlands programme. Crucial for these links are certain functions associated to automorphic forms, called L-functions, which are the subject of some of the most important conjectures of Mathematics.In recent years, investigations into the theory of automorphic forms have led into the study of variants of automorphic forms and of their L-functions, such as quasi-modular forms, harmonic Maass forms, mock modular forms, higher order modular forms and multiple Dirichlet series. In most cases, the motivation for introducing these objects was not just to generalize the classical automorphic forms and their L-functions, but to obtain novel tools to address already stated number theoretic problems. The techniques associated with these new objects in turn raise new interesting questions and highlight connections beyond the original motivating problems. For example, the theory of harmonic Maass forms and modular forms has been used to resolve problems in partitions of numbers, and higher order modular forms have been applied to Percolation Theory problems in Physics.As these techniques have only recently been discovered, they lead to a number of very interesting open questions, e.g. how to construct mock modular forms encoding specific partition functions, how to determine the arithmetic nature of high-order forms or how to use the theory of multiple Dirichlet series to bound moments of the Riemann zeta function. Questions of this type, are highly relevant both for the outstanding problems in classical automorphic forms and for the further development of the new subjects themselves. Therefore, many of these questions are very appropriate for a PhD project.