Translator Disclaimer
15 May 2014 Zeta functions of regular arithmetic schemes at s=0
Baptiste Morin
Duke Math. J. 163(7): 1263-1336 (15 May 2014). DOI: 10.1215/00127094-2681387

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.

Citation

Download Citation

Baptiste Morin. "Zeta functions of regular arithmetic schemes at s=0." Duke Math. J. 163 (7) 1263 - 1336, 15 May 2014. https://doi.org/10.1215/00127094-2681387

Information

Published: 15 May 2014
First available in Project Euclid: 9 May 2014

zbMATH: 06303878
MathSciNet: MR3205726
Digital Object Identifier: 10.1215/00127094-2681387

Subjects:
Primary: 11G40, 14F20

Rights: Copyright © 2014 Duke University Press

JOURNAL ARTICLE
74 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.163 • No. 7 • 15 May 2014
Back to Top