A number field analogue of Weil's theorem on congruent zeta functions

Abstract

Let $K$ be a function field of one variable over a finite field $\mathbb{F}$. Weil's celebrated theorem states that the congruent zeta function of $K/\mathbb{F}$ is determined by the $\mathrm{Gal}(\overline{\mathbb{F}}/\mathbb{F})$-module structure of $X_{\overline{\mathbb{F}}K}(p)\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$, and vise versa, where $p$ is a prime number different from the characteristic of $K$ and $X_{\overline{\mathbb{F}}K}(p)$ stands for the Galois group of the maximal unramified abelian $p$-extension over $\overline{\mathbb{F}}K$. In the present paper, I will give a number field analogue of the above mentioned theorem by considering the total cyclotomic extension, which we may regard as a number field analogue of $\overline{\mathbb{F}}K/K$.