Weil-\'etale cohomology and Zeta-values of proper regular arithmetic schemes

TL;DR

This paper proposes a conjectural framework linking Weil-étale cohomology to the special values of Zeta functions of proper regular arithmetic schemes, extending previous work and aligning with the Tamagawa number conjecture.

Contribution

It introduces a new conjecture relating Zeta function invariants to Weil-étale cohomology for a broad class of arithmetic schemes, generalizing prior results.

Findings

Conjecture aligns with Tamagawa number conjecture in known cases

Extends previous work on Zeta values and cohomology

Provides a unified framework for arithmetic schemes

Abstract

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $\mathcal{X}$ at any integer $n$ in terms of Weil-\'etale cohomology complexes. This extends work of Lichtenbaum \cite{Lichtenbaum05} and Geisser \cite{Geisser04b} for $\mathcal{X}$ of characteristic $p$, of Lichtenbaum \cite{li04} for $\mathcal{X}=\mathrm{Spec}(\mathcal{O}_F)$ and $n=0$ where $F$ is a number field, and of the second author for arbitrary $\mathcal{X}$ and $n=0$ \cite{Morin14}. We show that our conjecture is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou \cite{fpr91} if $\mathcal{X}$ is smooth over $\mathrm{Spec}(\mathcal{O}_F)$, and hence that it holds in cases where the Tamagawa number conjecture is known.