L-functions of automorphic forms and their variants

经典的自同构形式是处理困难的数论问题的有力工具。它们提供了研究算术问题的分析、代数和几何方面的联系，因此，它们是数论主要研究方案的核心，例如朗兰兹方案。对这些联系至关重要的是与自同构形式有关的某些函数，称为L函数，它们是一些最重要的数学猜想的主题。近年来，对自同构形式理论的研究导致了对自同构形式的变体及其L函数的研究，如拟模形式、调和Maas型、模拟模形式、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是为了推广经典的自同构形及其L函数，而且是为了获得新的工具来解决已经提出的数论问题。与这些新物体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形式和模形式理论被用来解决数的划分问题，高阶模形式被应用于物理中的渗流理论问题。由于这些技术是最近才被发现的，它们导致了一些非常有趣的开放问题，例如如何构造编码特定配分函数的模拟模形式，如何确定高阶形式的算术性质，如何利用多重Dirichlet级数理论来限制Riemann Zeta函数的矩。这类问题既与经典自同构形式中的突出问题密切相关，也与新学科本身的进一步发展密切相关。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.