The Étale Homotopy Type and Obstructions to the Local-Global Principle

Abstract

In 1969 Artin and Mazur defined the étale homotopy type of an algebraic variety \cite{AMa69}. In this paper we define various obstructions to the local-global principle on a variety $X$ over a global field using the étale homotopy type of $X$ and the concept of homotopy fixed points. We investigate relations between those "homotopy obstructions" and connect them to various known obstructions such as the Brauer -Manin obstruction, the étale-Brauer obstruction and finite descent obstructions. This gives a reinterpretation of known arithmetic obstructions in terms of homotopy theory.

This paper has been withdrawn by the author since the most recent version is in http://arxiv.org/abs/1110.0164

References in corpus (4)

• Twisted sheaves and the period-index problem
• Insufficiency of the Brauer-Manin obstruction applied to etale covers
• On the Brauer-Manin Obstruction Applied to Ramified Covers
• Continuous group actions on profinite spaces