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Zeta functions of regular arithmetic schemes at s=0

By Baptiste Morin

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.Comment: 53 pages. To appear in Duke Math.

Topics: Mathematics - Number Theory, Mathematics - Algebraic Geometry, 14F20, 14G10, 11S40, 11G40, 19F27
Year: 2013
DOI identifier: 10.1215/00127094-2681387
OAI identifier: oai:arXiv.org:1103.6061
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