Zeta functions of regular arithmetic schemes at s=0

Details Zeta functions of regular arithmetic schemes at s=0 Zeta functions of regular arithmetic schemes at s=0

2011-03-30

arxiv.org/abs/1103.6061v1

Mathematics

Number Theory

41 pages, submitted

Scientific paper

Lichtenbaum conjectured in \cite{Lichtenbaum} 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 motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we compute (unconditionally) the right Weil-\'etale cohomology of number rings and projective spaces over number rings. We state a precise version of Lichtenbaum's conjecture, which expresses the vanishing order (resp. the special value) of the Zeta function $\zeta(\mathcal{X},s)$ at $s=0$ as the rank (resp. the determinant) of a single 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. Lichtenbaum's conjecture for projective spaces over the ring of integers of an abelian number field follows.

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