Algebraic structures on automorphic L-functions

Abstract: Considerthe function field $F$ of a smooth curve over $\mathbb{F}_q$, with $q\neq 2$.

L-functions of automorphic representations of $\GL(2)$over $F$ are important objects for studying the arithmetic properties of thefield $F$. Unfortunately, they can be defined in two different ways: one byGodement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that theresulting L-functions coincide using a complicated computation.

I will present a conceptual proof that the two familiescoincide, by categorifying the question. This correspondence will necessitatecomparing two very different sets of data, which will have significantimplications for the representation theory of $\GL(2)$. In particular, we willobtain an exotic symmetric monoidal structure on the category ofrepresentations of $\GL(2)$

algebraic geometryalgebraic topologycategory theoryrings and algebrasrepresentation theory

Audience: researchers in the topic

Seminar on Representation Theory and Algebraic Geometry