Degree and the Brauer–Manin obstruction

Creutz, Brendan; Viray, Bianca

doi:10.2140/ant.2018.12.2445

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Degree and the Brauer–Manin obstruction

Brendan Creutz and Bianca Viray

Appendix: Alexei N. Skorobogatov

Vol. 12 (2018), No. 10, 2445–2470

DOI: 10.2140/ant.2018.12.2445

Abstract

Let $X \subset ℙ_{k}^{n}$ be a smooth projective variety of degree $d$ over a number field $k$ 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. In the case of Kummer surfaces we show, more specifically, that the obstruction is already given by the $2$ -primary torsion, and indeed that this holds for higher-dimensional Kummer varieties as well. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.

Keywords

Brauer–Manin obstruction, degree, period, rational points

Mathematical Subject Classification 2010

Primary: 14G05

Secondary: 11G35, 14F22

Milestones

Received: 19 December 2017

Revised: 12 July 2018

Accepted: 23 August 2018

Published: 1 February 2019

Authors

Brendan Creutz
School of Mathematics and Statistics University of Canterbury Christchurch New Zealand

Bianca Viray
Department of Mathematics University of Washington Seattle, WA United States

Alexei N. Skorobogatov
Department of Mathematics Imperial College London South Kensington Campus United Kingdom	Institute for the Information Transmission Problems Russian Academy of Sciences Moscow Russia

Vol. 12, No. 10, 2018