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

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.