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.