Algebraic Structures on Automorphic L-Functions
Consider the function field F of a smooth curve over F_q, with q > 2.
L-functions of automorphic representations of GL(2) over F are important objects for studying the arithmetic properties of the field F. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation.
I will present a conceptual proof that the two families coincide, by categorifying the question. This correspondence will necessitate comparing two very different sets of data, which will have significant implications for the representation theory of GL(2). In particular, we will obtain an exotic symmetric monoidal structure on the category of representations of GL(2).
No prior knowledge of automorphic forms is assumed. This work is a part of my PhD thesis under the supervision of J. Bernstein.