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

Abstract

Abstract 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 expression of 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 [4].