Zeta functions of regular arithmetic schemes at s=0

Abstract

Lichtenbaum conjectured the existence of a Weil-étale 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–Poincaré characteristics. Assuming the (conjectured) finite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $Spec\left(\mathbb{Z}\right)$. In particular, we obtain (unconditionally) the right Weil-étale 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 Soulé’s conjecture and to the Tamagawa number conjecture of Bloch–Kato, and deduce its validity in simple cases.