Insufficiency of the Brauer-Manin obstruction applied to etale covers

TL;DR

This paper constructs a specific variety over a global field where the failure of having rational points cannot be explained by the Brauer-Manin obstruction, even when considering finite etale covers, challenging existing explanations.

Contribution

It provides a counterexample demonstrating the insufficiency of the Brauer-Manin obstruction and its finite etale cover extension in explaining the absence of rational points.

Findings

Constructed a variety with no rational points not explained by Brauer-Manin obstruction

Shows limitations of Brauer-Manin obstruction in explaining rational point failures

Challenges the completeness of known obstructions in arithmetic geometry

Abstract

Let k be any global field of characteristic not 2. We construct a k-variety X such that X(k) is empty, but for which the emptiness cannot be explained by the Brauer-Manin obstruction or even by the Brauer-Manin obstruction applied to finite etale covers.