论文标题
$ \ mathrm {gl}（1）$的Zeta积分模块
Modules of Zeta Integrals for $\mathrm{GL}(1)$
论文作者
论文摘要
我们通过用“ Zeta积分模块”替换L功能来对$ \ mathrm {gl}（1）$的Hecke L功能进行分类。这些Zeta积分的模块是由经典L功能生成的。这种方法使我们可以通过避免通常用于定义它们的GCD程序来对有关L功能的问题进行分类，并使其构造更具规范性。 令$ f $为一个数字字段，让$ \ mathfrak {x}（f）$为$ f $上的hecke字符的空间。我们在$ \ mathfrak {x}（f）$和$ \ mathscr {s} $ - module $ \ mathscr {l} $ zeta Integrals上的$ \ mathfrak {x}（x} $。在某些本地化中，使用$ \ Mathscr {s}'$ of $ \ Mathscr {s} $中的行捆绑包$ \ Mathscr {l} $的规范琐事，我们表明Zeta Integrals $ \ Mathscr {l} $的模块包含与hecke l-funuction相同的信息，在这里，l功能被认为是$ \ mathfrak {x}（f）$上的单个函数。 在论文Arxiv：2011.03313中，作者已经隐式地使用了这一想法来推导自动形式理论的非平凡应用。本文的目的使“ Zeta积分模块”的概念明确而严格，以实现未来的应用。 最终结果与Connes和Meyer的结构密切相关，尽管从另一种角度看。 本文基于作者论文的一部分。
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.
