Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes

Abstract

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme \mathcal{X} at any integer n in terms of Weil-étale cohomology complexes. This extends work of S. Lichtenbaum [Compos. Math. 141, No. 3, 689–702 (2005; Zbl 1073.14024)] and T. Geisser [Math. Ann. 330, No. 4, 665–692 (2004; Zbl 1069.14021)] for \mathcal{X} of characteristic p , of S. Lichtenbaum [Ann. Math. (2) 170, No. 2, 657–683 (2009; Zbl 1278.14029)] for \mathcal{X}=\mathrm{Spec}(\mathcal{O}_F) and n=0 where F is a number field, and of the second author for arbitrary \mathcal{X} and n=0 [B. Morin, Duke Math. J. 163, No. 7, 1263–1336 (2014; Zbl 06303878)]. We show that our conjecture is compatible with the Tamagawa number conjecture of S. Bloch and K. Kato [Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)], and J.-M. Fontaine and B. Perrin-Riou [Proc. Symp. Pure Math. 55, 599–706 (1994; Zbl 0821.14013)] if \mathcal{X} is smooth over \mathrm{Spec}(\mathcal{O}_F) , and hence that it holds in cases where the Tamagawa number conjecture is known.