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arXiv · 1703.02187

Degree and the Brauer-Manin obstruction

Brendan Creutz·Bianca Viray
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Abstract

Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show that the obstruction is already given by the 2-primary torsion. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.

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Brendan Creutz, Bianca Viray. 2018-07-23. Degree and the Brauer-Manin obstruction. https://doi.org/10.2140/ant.2018.12.2445

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