Homotopy Obstructions to Rational Points

Abstract

In this paper we propose to use a relative variant of the notion of the \'{e}tale homotopy type of an algebraic variety in order to study the existence of rational points on it. In particular, we use an appropriate notion of homotopy fixed points in order to construct obstructions to the local-global principle. The main results in this paper are the connections between these obstructions and the classical obstructions, such as the Brauer-Manin, the \'{e}tale-Brauer and certain descent obstructions. These connections allow one to understand the various classical obstructions in a unified framework.