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 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(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 ζ(X,s) at s=0 in terms of a perfect complex of abelian groups RΓW,c(X,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.