Monoidal Structures on GL(2)-Modules and Abstractly Automorphic Representations

Consider the function field F of a smooth curve over FqFq, with q>2q>2.

L-functions of automorphic representations of GL(2)GL(2) over FF are important objects for studying the arithmetic properties of the field FF. 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 GL(2)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.