Insufficiency of the Brauer–Manin Obstruction for Rational Points on Enriques Surfaces

In Varilly-Alvarado and Viray (Adv. Math. 226(6):4884–4901, 2011), the authors 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 \(X_{\overline{\mathbb{Q}}}\) does not descend to \(\mathbb{Q}\). Together with the results of Varilly-Alvarado and Viray (Adv. Math. 226(6):4884–4901, 2011), this proves that the Brauer–Manin obstruction is insufficient to explain all failures of the Hasse principle on Enriques surfaces.