Duke Mathematical Journal

Zeta functions of regular arithmetic schemes at s=0

Baptiste Morin

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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.

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Duke Math. J., Volume 163, Number 7 (2014), 1263-1336.

First available in Project Euclid: 9 May 2014

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Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]


Morin, Baptiste. Zeta functions of regular arithmetic schemes at $s=0$. Duke Math. J. 163 (2014), no. 7, 1263--1336. doi:10.1215/00127094-2681387. https://projecteuclid.org/euclid.dmj/1399645816

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