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

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

2010-02-07

arxiv.org/abs/1002.1423v4

Mathematics

Algebraic Geometry

This paper has been withdrawn by the author since the most recent version is in http://arxiv.org/abs/1110.0164

Scientific paper

In 1969 Artin and Mazur defined the \'etale 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 \'etale 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 \'etale-Brauer obstruction and finite descent obstructions. This gives a reinterpretation of known arithmetic obstructions in terms of homotopy theory.

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