Abstract
It had long been suspected that varieties X over a number field k without k-points but with a non-empty Brauer–Manin set X (Ak) are fairly common. The first examples were found in [27] and then in [2]. An earlier, conditional example is given in [21]. One should also expect that there are many varieties X without k-points for which the etale Brauer–Manin set X (Ak) ⊂ X (Ak) is non-empty (we refer to [19] or [29] for the definition of these subsets of the space X (Ak) of adeles of X ). Different methods to construct such varieties have been found recently. In [19] Poonen constructs a threefold X with a surjective morphism to a curve C that has exactly one k-point P and the fibre XP has points everywhere locally but not globally. In Poonen’s example XP is a smooth Châtelet surface. The trick with a curve with just one rational point was also used in [11] where the fibres of X → C are curves of high genus and XP is a singular curve which geometrically is a union of projective lines. In retrospect one could note that the examples in [27] and [2] are families of genus 1 curves parameterised by elliptic curves of Mordell–Weil rank 0. In this paper we propose more flexible methods to construct such examples. We show that the varieties X such that X (k) = ∅ and X (Ak) = ∅ include the following:
Highlights
It had long been suspected that varieties X over a number field k without k-points but with a non-empty Brauer–Manin set X (Ak )Br are fairly common
A conic bundle surface X → E over a real quadratic field k, where E is an elliptic curve such that E(k) = {0}, see Sect. 5.2; a threefold over an arbitrary real number field k ⊂ R, which is a family X → C of 2-dimensional quadrics parameterised by a curve C with exactly one k-point, see Sect. 3.1; a threefold over an arbitrary number field k, which is a family X → C of geometrically rational surfaces parameterised by a curve C with exactly one k-point, the fibre above which is singular, see Sect. 3.2
In the first and second examples, in contrast to those previously known, the smooth fibres satisfy the Hasse principle and weak approximation. To put this into a historical perspective let us note that soon after Manin [17] introduced the obstruction bearing his name, Iskovskikh [13] constructed a counterexample to the Hasse principle on a conic bundle over the projective line over Q
Summary
It had long been suspected that varieties X over a number field k without k-points but with a non-empty Brauer–Manin set X (Ak )Br are fairly common. 3.1; a threefold over an arbitrary number field k, which is a family X → C of geometrically rational surfaces parameterised by a curve C with exactly one k-point, the fibre above which is singular, see Sect. In the first and second examples, in contrast to those previously known, the smooth fibres satisfy the Hasse principle and weak approximation To put this into a historical perspective let us note that soon after Manin [17] introduced the obstruction bearing his name, Iskovskikh [13] constructed a counterexample to the Hasse principle on a conic bundle over the projective line over Q. 4 we establish some Bashmakov-style properties of elliptic curves with a large Galois image on torsion points These properties are used in the proof of our main result in the case of surfaces in Sect. In April 2013 when the authors were guests of the Hausdorff Institut für Mathematik (Bonn) during the special programme “Arithmetic and geometry”
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