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The Hardy–Littlewood conjecture and rational points

Part of: Diophantine equations Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  10 September 2014

Yonatan Harpaz ,
Alexei N. Skorobogatov  and
Olivier Wittenberg
Yonatan Harpaz
Affiliation:
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands email harpazy@gmail.com
Alexei N. Skorobogatov
Affiliation:
Department of Mathematics, Imperial College London, SW7 2BZ, UK Institute for the Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow 127994, Russia email a.skorobogatov@imperial.ac.uk
Olivier Wittenberg
Affiliation:
Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France email wittenberg@dma.ens.fr
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Abstract

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Schinzel’s Hypothesis (H) was used by Colliot-Thélène and Sansuc, and later by Serre, Swinnerton-Dyer and others, to prove that the Brauer–Manin obstruction controls the Hasse principle and weak approximation on pencils of conics and similar varieties. We show that when the ground field is $\mathbb{Q}$ and the degenerate geometric fibres of the pencil are all defined over $\mathbb{Q}$, one can use this method to obtain unconditional results by replacing Hypothesis (H) with the finite complexity case of the generalised Hardy–Littlewood conjecture recently established by Green, Tao and Ziegler.

Keywords

rational pointsfibration methodBrauer–Manin obstructiongeneralised Hardy–Littlewood conjecture

MSC classification

Primary: 14G05: Rational points 11D57: Multiplicative and norm form equations
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 12 , December 2014 , pp. 2095 - 2111
Copyright
© The Author(s) 2014 

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