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 $\mathrm{Spec}(\mathbb{Z})$. 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.

53 pages. To appear in Duke Math. J

References in corpus (1)

• Semipurity of tempered Deligne cohomology

Cited by in corpus (2)

• Arakelov motivic cohomology I
• Special values of $L$-functions on regular arithmetic schemes of dimension $1$