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Failure of the Hasse principle for Enriques surfaces. (English) Zbl 1239.11068
Following Manin’s original work describing how the Brauer group of a variety can obstruct the existence of rational points, his construction (known as the Brauer–Manin obstruction) has been generalised in various ways to give several different obstructions. Those considered in this article, in increasing order of strength, are: the algebraic Brauer–Manin obstruction; the Brauer–Manin obstruction; and the Brauer–Manin obstruction applied to finite étale torsors, known as the étale-Brauer obstruction. Various authors have constructed examples in which these obstructions give genuinely different information about the rational points.
In this well-explained article, the authors fill a gap in the literature by constructing an Enriques surface $$X$$ over the rational numbers on which the étale-Brauer obstruction shows that there are no rational points, yet there is no algebraic Brauer–Manin obstruction to the Hasse principle. This leaves open the possibility that the absence of rational points is explained by the (necessarily transcendental) Brauer–Manin obstruction; the authors relate this to the corresponding question for a $$K3$$ surface. In the notation of the article:
Theorem 1.1. There exists an Enriques surface $$X/\mathbb Q$$ such that $X(\mathbb A_\mathbb Q)^{\text{ét},\mathbf{Br}} = \emptyset \qquad \text{and} \qquad X(\mathbb A_\mathbb Q)^{\mathbf{Br}_1} \neq \emptyset.$ Moreover, if $$X(\mathbb A_{\mathbb Q})^{\mathbf{Br}} = \emptyset$$, then $$Y(\mathbb A_\mathbb Q)^{\mathbf{Br}\, Y \setminus \mathbf{Br}_1 Y} = \emptyset$$, where $$Y$$ is a $$K3$$ double cover of $$X$$.
The variety $$X$$ in question is defined as the quotient of a $$K3$$ surface $$Y$$ by a fixed-point free involution; $$Y$$ itself is explicitly given as the intersection of three quadrics in $${\mathbb P}^5$$, parametrised by three positive integers $$a,b,c$$ subject to certain arithmetic conditions (described in Theorem 1.2). In particular, $$(a,b,c)=(12,111,13)$$ satisfies the conditions.
Since $$Y$$ is simply connected, the only finite étale torsors over $$X$$ are twists of $$Y \to X$$. To show that there is an étale-Brauer obstruction to the existence of rational points on $$X$$, it is enough to show that there is a Brauer–Manin obstruction to the existence of rational points on every twist of $$Y$$; the construction of $$Y$$ ensures that this comes down to a known example of a Brauer–Manin obstruction, given by B. J. Birch and H. P. F. Swinnerton-Dyer [J. Reine Angew. Math. 274–275, 164–174 (1975; Zbl 0326.14007)].
Proving the absence of an algebraic Brauer–Manin obstruction to rational points on $$X$$ is more intricate, and involves computing $$\mathbf{Pic} (\bar{X})$$ and its Galois cohomology. This is accomplished using a fibration of $$Y$$ into genus 1 curves; explicit equations for divisors, and the Galois action on them, are given in appendices.

##### MSC:
 11G35 Varieties over global fields 14G05 Rational points 14G25 Global ground fields in algebraic geometry 14F22 Brauer groups of schemes 14J28 $$K3$$ surfaces and Enriques surfaces
Magma
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