The Weil-\'etale fundamental group of a number field I

TL;DR

This paper explores a conjectural Grothendieck topology for arithmetic schemes, establishing a fundamental group linked to the Arakelov Picard group, and proposes topological properties aligning with Weil group axioms to compute Lichtenbaum cohomology.

Contribution

It introduces a fundamental group for a conjectural site of number rings, connecting it to the Arakelov Picard group and proposing axioms for a Weil group-inspired topology.

Findings

Fundamental group of the conjectural site is isomorphic to the Arakelov Picard group.

Proposed topological properties mirror Weil group axioms.

Topos satisfying these properties computes Lichtenbaum cohomology.

Abstract

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.