Automorphic forms: arithmetic and analytic interfaces

The goal of this project is to better understand the properties of automorphic forms of higher rank, especially in situations involving high ramification. Automorphic forms are central objects in the Langlands program, a vast web of theorems and conjectures that connects concepts coming from number theory, representation theory and geometry. The simplest examples of automorphic forms include Dirichlet characters and classical modular forms, both of which have proved to be of profound importance in modern mathematics. More generally, automorphic forms are complex valued functions that can be naturally viewed as vectors inside representations known as automorphic representations. This viewpoint allows one to associate automorphic forms to any reductive algebraic group. From a different point of view, automorphic forms include (as special cases) eigenfunctions of Laplacians on arithmetic manifolds. This viewpoint allows one to bring in a whole range of additional perspectives coming from analysis, spectral theory and quantum mechanics. Automorphic forms and the L functions attached to them have been key ingredients in the solutions of many famous and difficult problems, such as Wiles' proof of Fermat's last Theorem and Duke's work on the representations of algebraic integers by ternary quadratic forms.
A central theme in modern number theory is to understand key properties of automorphic forms and their associated L functions as one or more of their defining parameters vary. The finite or non archimedean part of these parameters can be captured by a fundamental arithmetic quantity called the conductor or level (henceforth denoted by N) that measures its total ramification (or complexity at finite primes). The level appears in the functional
equation of the attached L function, as well as (essentially) describes the arithmetic manifold that the automorphic form lives on. Compared to the archimedean aspect, there has been relatively little progress in the level aspect versions of analytic problems about automorphic forms, especially in higher rank. In this project, the student will investigate key questions related to the themes described above. The tools used will be a mix of algebraic as well as analytic number theory, together with representation theory of p adic groups. The specific problems to be solved will depend on the interests of the student (possible examples include sup norms and other L^p norms,
period formulas, etc.)