Exotic Monoidal Structures and Abstractly Automorphic Representations for $\mathrm{GL}(2)$

We use the theta correspondence to study the equivalence between Godement-Jacquet and Jacquet-Langlands L-functions for $\mathrm{GL}(2)$. We show that the resulting comparison is in fact an exotic symmetric monoidal structure on the category of $\mathrm{GL}(2)$-modules. Moreover, this enables us to construct an Abelian category of abstractly automorphic representations, whose irreducible objects are the usual automorphic representations. We speculate that this category is a natural setting for the study of automorphic phenomena for $\mathrm{GL}(2)$, and demonstrate its basic properties. This paper is a part of the author's thesis.