Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published March 2011 | public
Journal Article

The Weil-étale fundamental group of a number field I

Abstract

Lichtenbaum has conjectured (Ann of Math. (2) 170(2) (2009), 657-683) the existence of a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of the constant sheaf Z with compact support at infinity gives, up to sign, the leading term of the zeta function ζX(s) at s = 0. In this paper we consider the category of sheaves XL on this conjectural site for X(overbar) = Spec(Ο_F) the spectrum of a number ring. We show that X(overbar)_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 XL. 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 étale sheaves computing the expected Lichtenbaum cohomology.

Additional Information

© 2011 Faculty of Mathematics, Kyushu University. Received 28 July 2010. I am very grateful to Matthias Flach, Masanori Morishita and Frédéric Paugam for their comments.

Additional details

Created:
August 19, 2023
Modified:
October 23, 2023