Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers

Abstract

Following the ideas of Flach and Morin (2018), we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function $\zeta (X,s)$ at $s=n<0$, where $X$ is a separated scheme of finite type over $\mathrm{Spec}\mathbb{Z}$. We prove that the conjecture is compatible with closed-open decompositions of schemes and with affine bundles, and consequently, that it holds for cellular schemes over certain one-dimensional bases.