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

### Matthias Flach

Department of Mathematics, Caltech Pasadena, CA 91125, USA### Baptiste Morin

CNRS, IMB, Université de Bordeaux 351, cours de la Libération, F 33405 Talence cedex, France

## Abstract

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $X$ 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 $X$ of characteristic $p$, of S. Lichtenbaum [Ann. Math. (2) 170, No. 2, 657–683 (2009; Zbl 1278.14029)] for $X=Spec(O_{F})$ and $n=0$ where $F$ is a number field, and of the second author for arbitrary $X$ 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 $X$ is smooth over $Spec(O_{F})$, and hence that it holds in cases where the Tamagawa number conjecture is known.

## Cite this article

Matthias Flach, Baptiste Morin, Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes. Doc. Math. 23 (2018), pp. 1425–1560

DOI 10.4171/DM/651