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Morin, Baptiste

Zeta functions of regular arithmetic schemes at \(s=0\). (English) Zbl 1408.14076

Duke Math. J. 163, No. 7, 1263-1336 (2014).
Summary: 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 \(Spec(\mathbb Z)\). 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 \({\zeta}(X,s)\) at \(s=0\) in terms of a perfect complex of abelian groups \(R{\Gamma}W,c(X,Z)\). 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.

Cited in 1 Review
Cited in 3 Documents

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14F42 Motivic cohomology; motivic homotopy theory
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\textit{B. Morin}, Duke Math. J. 163, No. 7, 1263--1336 (2014; Zbl 1408.14076)
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