Weil-\'etale cohomology and zeta-values of arithmetic schemes at negative integers

TL;DR

This paper proposes a conjecture relating Weil-étale cohomology to the special values of zeta functions at negative integers for arithmetic schemes, and verifies its consistency with certain scheme decompositions and classes of schemes.

Contribution

It formulates a conjecture connecting Weil-étale cohomology with zeta-values at negative integers and proves its compatibility with scheme decompositions and specific classes of schemes.

Findings

Conjecture aligns Weil-étale cohomology with zeta-values at negative integers.

Compatibility established with closed-open decompositions and affine bundles.

Confirmed for cellular schemes over certain one-dimensional bases.

Abstract

Following the ideas of Flach and Morin (Doc. Math. 23 (2018), 1425--1560), we state a conjecture in terms of Weil-\'etale 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 $\operatorname{Spec} \mathbb{Z}$. 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. This is a continuation of arXiv:2012.11034, which gives a construction of Weil-\'etale cohomology for $n < 0$ under the mentioned assumptions on $X$.