Algebra & Number Theory

Degree and the Brauer–Manin obstruction

Brendan Creutz and Bianca Viray

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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.

Article information

Source
Algebra Number Theory, Volume 12, Number 10 (2018), 2445-2470.

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1550113228

Digital Object Identifier
doi:10.2140/ant.2018.12.2445

Mathematical Reviews number (MathSciNet)
MR3911136

Zentralblatt MATH identifier
07026823

Subjects
Primary: 14G05: Rational points
Secondary: 11G35: Varieties over global fields [See also 14G25] 14F22: Brauer groups of schemes [See also 12G05, 16K50]

Keywords
Brauer–Manin obstruction degree period rational points

Citation

Creutz, Brendan; Viray, Bianca. Degree and the Brauer–Manin obstruction. Algebra Number Theory 12 (2018), no. 10, 2445--2470. doi:10.2140/ant.2018.12.2445. https://projecteuclid.org/euclid.ant/1550113228


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