Insufficiency of the étale Brauer-Manin obstruction: towards a simply connected example

Abstract

Since Poonen's construction of a variety $X$ defined over a number field $k$ for which $X(k)$ is empty and the étale Brauer--Manin set $X({\mathbf{A}}_k{)}^{\text{Br,et}}$ is not, several other examples of smooth, projective varieties have been found for which the étale Brauer--Manin obstruction does not explain the failure of the Hasse principle. All known examples are constructed using "Poonen's trick", i.e. they have the distinctive feature of being fibrations over a higher genus curve; in particular, their Albanese variety is non-trivial. In this paper, we construct examples for which the Albanese variety is trivial. The new geometric ingredient in our construction is the appearance of Beauville surfaces. Assuming the $abc$ conjecture and using geometric work of Campana on orbifolds, we also prove the existence of an example which is simply connected.

13 pages; final version, to appear in American Journal of Mathematics

References in corpus (2)

• Birational geometry for number theorists
• Pathologies of the Brauer-Manin obstruction