Insufficiency of the Brauer-Manin obstruction for Enriques surfaces

In 2011, V\`arilly-Alvarado and the last author constructed an Enriques surface $X$ over $\mathbb{Q}$ with an \'etale-Brauer obstruction to the Hasse principle and no algebraic Brauer-Manin obstruction. In this paper, we show that the nontrivial Brauer class of $X_{\bar{\mathbb{Q}}}$ does not descend to $\mathbb{Q}$. Together with the results of V\`arilly-Alvarado and the last author, this proves that the Brauer-Manin obstruction is insufficient to explain all failures of the Hasse principle on Enriques surfaces. The methods of this paper build on the ideas in several recent papers by the last author and various collaborators: we study geometrically unramified Brauer classes on $X$ via pullback of ramified Brauer classes on a rational surface. Notably, we develop techniques which work over fields which are not necessarily separably closed, in particular, over number fields.