Modules of Zeta Integrals for $\mathrm{GL}(1)$

TL;DR

This paper categorifies Hecke L-functions for GL(1) by introducing modules of zeta integrals, providing a more canonical and explicit framework that encapsulates L-functions as modules over a ring of holomorphic functions.

Contribution

It formalizes the concept of modules of zeta integrals for GL(1), making their construction explicit and canonical, and relates them to existing frameworks like Connes and Meyer's approach.

Findings

Modules of zeta integrals encode the same information as Hecke L-functions.

The approach avoids GCD procedures, making the construction more canonical.

Connections to automorphic representation theory and existing frameworks.

Abstract

We categorify the Hecke L-functions of $\mathrm{GL}(1)$ by replacing the L-functions with "modules of zeta integrals". These modules of zeta integrals are generated by the classical L-function. This approach allows us to categorify questions regarding L-functions, as well as make their construction more canonical by avoiding the GCD procedure usually used to define them. Let $F$ be a number field, and let $\mathfrak{X}(F)$ be the space of Hecke characters on $F$. We define a ring $\mathscr{S}$ of holomorphic functions on $\mathfrak{X}(F)$ and an $\mathscr{S}$-module $\mathscr{L}$ of zeta integrals on $\mathfrak{X}(F)$. Using a canonical trivialization of the line bundle $\mathscr{L}$ in some localization $\mathscr{S}'$ of $\mathscr{S}$, we show that the module of zeta integrals $\mathscr{L}$ contains the same information as the Hecke L-function of $\mathrm{GL}(1)$. Here, the L-function is thought of as a single function on $\mathfrak{X}(F)$. In the paper arXiv:2011.03313, the author has already implicitly used this idea to derive non-trivial applications for the theory of automorphic representations. The goal of this paper make the notion of "module of zeta integrals" explicit and rigorous for future applications. The end result is very closely related to a construction by Connes and Meyer, although seen from an alternative point of view. This paper is based on a part of the author's thesis.