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

Abstract

In Várilly-Alvarado and Viray (Adv. Math. 226(6):4884–4901, 2011), the authors constructed an Enriques surface X over $\mathbb{Q}$ with an étale-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 Várilly-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.

The methods of this paper build on the ideas in Creutz and Viray (Math. Ann. 362(3–4):1169–1200, 2015; Manuscripta Math. 147(1–2): 139–167, 2015) and Ingalls et al., (Unramified Brauer classes on cyclic covers of the projective plane, Preprint): 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.

Appendix: Fields, Defining Equations, and Galois Actions

The splitting field K of the genus 1 curves $C_{i},\widetilde{C}_{i},\widetilde{D}_{i},F_{i},G_{i}$ is generated by

$$\displaystyle\begin{array}{rcl} & & i,\sqrt{2},\sqrt{5},\sqrt{a},\sqrt{c},\eta _{0}:= \sqrt{c^{2 } - 100ab},\gamma _{0}:= \sqrt{-c^{2 } - 5bc - 10ac - 25ab}, {}\\ & & \root{4}\of{ab},\sqrt{-2 + 2\sqrt{2}},\sqrt{-c - 10\sqrt{ab}}, {}\\ & & \theta _{0}:= \sqrt{4a^{2 } + b^{2}},\;\xi _{0}:= \sqrt{a + b + c/5},\;\xi _{0}':= \sqrt{a + b/4 + c/10}, {}\\ & & \theta _{1}^{+}:= \sqrt{20a^{2 } - 10ab - 2bc + (10a + 2c)\theta _{ 0}},\;\theta _{2}^{+}:= \sqrt{-5a - 5/2b - 5/2\theta _{ 0}}, {}\\ & & \xi _{1}^{+}:= \sqrt{20a + 10b + 3c + 20\xi _{ 0}\xi _{0}'},\;\xi _{2}^{+}:= \sqrt{4a + 2b + 2/5c + 4\xi _{ 0}\xi _{0}'}. {}\\ \end{array}$$

Define

$$\displaystyle{ \begin{array}{rclcrcl} \eta _{1}^{+} &:=& \frac{c-\eta _{0}+10\sqrt{ab}} {10\sqrt{a}\sqrt{-c-10\sqrt{ab}}} &&\gamma _{1}^{+} &:=&(\theta _{1}^{+})^{-1}\left (10a^{2} - 5ab - bc + 2a\gamma _{0} + (c + 5a)\theta _{0}\right ) \\ \eta _{1}^{-}&:=& \frac{c+\eta _{0}+10\sqrt{ab}} {10\sqrt{a}\sqrt{-c-10\sqrt{ab}}} &&\gamma _{1}^{-}&:=&(\theta _{1}^{+})^{-1}\left (10a^{2} - 5ab - bc - 2a\gamma _{0} + (c + 5a)\theta _{0}\right ). \\ \end{array} }$$

The following subfields of K are of particular interest:

$$\displaystyle{ \mathbb{Q}\; \subset \; K_{0}: =\mathbb{Q}\left (i,\sqrt{2},\sqrt{5},\sqrt{-2+ 2\sqrt{2}}\right )\; \subset \; K_{1}: =K_{0}\left (\theta _{0},\sqrt{ab},\eta _{1}^{+},\gamma _{ 1}^{+}\right )\; \subset \; K. }$$

The field extensions $K/\mathbb{Q}$ and $K_{1}/\mathbb{Q}$ are Galois, as the following relations show:

$$\displaystyle\begin{array}{rcl} \sqrt{-2 - 2\sqrt{2}}& =& \frac{2i} {\sqrt{-2 + 2\sqrt{2}}}, {}\\ \sqrt{ -c + 10\sqrt{ab}}& =& \frac{\eta _{0}} {\sqrt{-c - 10\sqrt{ab}}}, {}\\ \theta _{1}^{-}&:=& \sqrt{20a^{2 } - 10ab - 2bc + (10a + 2c)\theta _{ 0}} = \frac{4a\gamma _{0}} {\theta _{1}^{+}}, {}\\ \theta _{2}^{-}&:=& \sqrt{-5a - 5/2b + 5/2\theta _{ 0}} = \frac{5\sqrt{ab}} {\theta _{2}^{+}}, {}\\ \xi _{1}^{-}&:=& \sqrt{20a + 10b + 3c - 20\xi _{ 0}\xi _{0}'} = \frac{\eta _{0}} {\xi _{1}^{+}}, {}\\ \xi _{2}^{-}&:=& \sqrt{4a + 2b + 2/5c - 4\xi _{ 0}\xi _{0}'} = \frac{2\gamma _{0}} {5\xi _{2}^{+}}, {}\\ (\gamma _{1}^{+})^{2}& =& 10a^{2} - 5ab - bc + 2a\gamma _{ 0},\quad (\eta _{1}^{+})^{2} = \frac{-c +\eta _{0}} {50a}, {}\\ (\gamma _{1}^{-})^{2}& =& 10a^{2} - 5ab - bc - 2a\gamma _{ 0},\quad (\eta _{1}^{-})^{2} = \frac{-c -\eta _{0}} {50a}, {}\\ \gamma _{1}^{+}\gamma _{ 1}^{-}& =& (5a + c)\theta _{ 0},\quad \eta _{1}^{+}\eta _{ 1}^{-} = \frac{-1} {5a} \sqrt{ab}. {}\\ \end{array}$$

Tables A.1 and A.4 show that these fields have the properties claimed in Sect. 3.

Remark 1.

Tables A.1 and A.4 list defining equations of a subvariety of the Y _a . The image of this subvariety gives the corresponding object in X _a (Table A.5).

Table A.4 Defining equations of a curve representing a divisor class

Table A.5 The Galois action on the fibers of the genus 1 fibrations and the Weierstrass points