Duke Mathematical Journal

Zeta functions of regular arithmetic schemes at s=0

Baptiste Morin

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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.

Article information

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

Dates
First available in Project Euclid: 9 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1399645816

Digital Object Identifier
doi:10.1215/00127094-2681387

Mathematical Reviews number (MathSciNet)
MR3205726

Zentralblatt MATH identifier
06303878

Subjects
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]

Citation

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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