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2018 Degree and the Brauer–Manin obstruction
Brendan Creutz, Bianca Viray
Algebra Number Theory 12(10): 2445-2470 (2018). DOI: 10.2140/ant.2018.12.2445

Abstract

Let Xkn 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.

Citation

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Brendan Creutz. Bianca Viray. "Degree and the Brauer–Manin obstruction." Algebra Number Theory 12 (10) 2445 - 2470, 2018. https://doi.org/10.2140/ant.2018.12.2445

Information

Received: 19 December 2017; Revised: 12 July 2018; Accepted: 23 August 2018; Published: 2018
First available in Project Euclid: 14 February 2019

zbMATH: 07026823
MathSciNet: MR3911136
Digital Object Identifier: 10.2140/ant.2018.12.2445

Subjects:
Primary: 14G05
Secondary: 11G35, 14F22

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.12 • No. 10 • 2018
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