Insufficiency of the Brauer-Manin obstruction applied to étale covers

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 étale covers.