L-functions of automorphic forms and their variants

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.