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

L-functions of automorphic representations of $\mathrm{GL}(2)$ 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.

Each of these L-functions is the GCD of a family of zeta integrals associated to test data. I will categorify the question, by showing that there is a correspondence between the two families of zeta integrals, instead of just their L-functions. The resulting comparison of test data will induce an exotic symmetric monoidal structure on the category of representations of $\mathrm{GL}(2)$.

It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.