The Weil-étale fundamental group of a number field I

Details The Weil-étale fundamental group of a number field I The Weil-étale fundamental group of a number field I

2010-06-02

arxiv.org/abs/1006.0523v2

Mathematics

Number Theory

40 pages. To appear in Kyushu Journal of Mathematics

Scientific paper

Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to sign, the leading term of the zeta-function $\zeta_X(s)$ at $s=0$. In this paper we consider the category of sheaves $\bar{X}_L$ on this conjectural site for $X=Spec(\mathcal{O}_F)$ the spectrum of a number ring. We show that $\bar{X}_L$ has, under natural topological assumptions, a well defined fundamental group whose abelianization is isomorphic, as a topological group, to the Arakelov Picard group of $F$. This leads us to give a list of topological properties that should be satisfied by $\bar{X}_L$. These properties can be seen as a global version of the axioms for the Weil group. Finally, we show that any topos satisfying these properties gives rise to complexes of \'etale sheaves computing the expected Lichtenbaum cohomology.

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