Homotopy Obstructions to Rational Points

TL;DR

This paper introduces a homotopy-theoretic framework to analyze rational points on algebraic varieties, connecting classical obstructions like Brauer-Manin with new homotopy-based obstructions.

Contribution

It develops a relative tale homotopy type approach and links it to classical obstructions, providing a unified perspective on local-global principles.

Findings

Homotopy obstructions relate to classical obstructions such as Brauer-Manin.

Connections established between homotopy fixed points and classical obstructions.

Framework unifies various obstructions under a homotopy-theoretic approach.

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.