Zeta-values of one-dimensional arithmetic schemes at strictly negative integers

TL;DR

This paper derives a formula for the special values of zeta functions of one-dimensional arithmetic schemes at negative integers, linking them to étale motivic cohomology and regulators, and proves it under certain abelian extension conditions.

Contribution

The paper provides a new explicit formula for zeta values at negative integers for one-dimensional schemes, extending previous Weil-étale formalism and conjecturing its general validity.

Findings

Formula for zeta values in terms of motivic cohomology and regulators

Proof of the formula under abelian extension assumptions

Calculation of Weil-étale cohomology for these schemes

Abstract

Let $X$ be an arithmetic scheme (i.e., separated, of finite type over $\operatorname{Spec} \mathbb{Z}$) of Krull dimension $1$. For the associated zeta function $\zeta (X,s)$, we write down a formula for the special value at $s = n < 0$ in terms of the \'{e}tale motivic cohomology of $X$ and a regulator. We prove it in the case when for each generic point $\eta \in X$ with $\operatorname{char} \kappa (\eta) = 0$, the extension $\kappa (\eta)/\mathbb{Q}$ is abelian. We conjecture that the formula holds for any one-dimensional arithmetic scheme. This is a consequence of the Weil-\'{e}tale formalism developed by the author in [arXiv:2012.11034] and [arXiv:2102.12114], following the work of Flach and Morin (Doc. Math. 23 (2018), 1425--1560). We also calculate the Weil-\'{e}tale cohomology of one-dimensional arithmetic schemes and show that our special value formula is a particular case of the main conjecture from [arXiv:2102.12114].