The \'{E}tale Homotopy Type and Obstructions to the Local-Global Principle

TL;DR

This paper explores how the étale homotopy type of algebraic varieties can be used to define and relate various obstructions to the local-global principle, connecting homotopy theory with arithmetic obstructions.

Contribution

It introduces homotopy-theoretic obstructions to the local-global principle and relates them to classical obstructions like the Brauer-Manin and étale-Brauer obstructions.

Findings

Homotopy obstructions generalize classical arithmetic obstructions.

Connections established between homotopy fixed points and known obstructions.

Reinterpretation of arithmetic obstructions via homotopy theory.

Abstract

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.