Vol. 12, No. 10, 2018

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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

Let X kn 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.

Brauer–Manin obstruction, degree, period, rational points
Mathematical Subject Classification 2010
Primary: 14G05
Secondary: 11G35, 14F22
Received: 19 December 2017
Revised: 12 July 2018
Accepted: 23 August 2018
Published: 1 February 2019
Brendan Creutz
School of Mathematics and Statistics
University of Canterbury
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