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

© 2017 by Johns Hopkins University Press. 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(Ak)Br,ét 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.

Publication year:2017

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BOF-publication weight:6

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Authors from:Higher Education

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