Zeta functions of regular arithmetic schemes at s=0

TL;DR

This paper constructs a Weil-étale cohomology theory for regular arithmetic schemes, aiming to describe the Zeta function's behavior at s=0 and relating it to existing conjectures in number theory.

Contribution

It develops a cohomology framework for regular schemes over Spec(Z), connecting Zeta function properties to conjectures like Soulé's and Bloch-Kato's.

Findings

Constructed Weil-étale cohomology for regular schemes over number rings.

Formulated a conjecture linking Zeta function values to cohomology complexes.

Validated the conjecture in specific simple cases.

Abstract

Lichtenbaum conjectured the existence of a Weil-\'etale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler-Poincar\'e characteristics. Assuming the (conjectured) finite generation of some \'etale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we obtain (unconditionally) the right Weil-\'etale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing order and the special value up to sign of the Zeta function $\zeta(\mathcal{X},s)$ at $s=0$ in terms of a perfect complex of abelian groups $R\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$. Then we relate this conjecture to Soul\'e's conjecture and to the Tamagawa number conjecture of Bloch-Kato, and deduce its validity in simple cases.