Student Number Theory: Automorphic forms and their applications

In this talk, I'll explain how to extend the classical theory of modular forms to the theory of automorphic forms and automorphic representations, with a focus on GL2. I'll give an overview of how the local theory yields the global theory (on the adeles) and will explain some of the advances of Jacquet-Langlands. Finally, I'll describe some of the important tools that arise in modern research on automorphic forms.