The Hardy--Littlewood conjecture and rational points

Abstract

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 Q and the degenerate geometric fibres of the pencil are all defined over Q, one can use these methods 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.

19 pages; minor changes, final version

References in corpus (1)

• Norms as products of linear polynomials

Cited by in corpus (7)

• On the fibration method for zero-cycles and rational points
• Principes locaux-globaux pour certaines fibrations en torseurs sous un tore
• Schinzel Hypothesis on average and rational points
• Rational points on fibrations with few non-split fibres
• Sieves and the Minimal Ramification Problem
• Strong approximation for a family of norm varieties
• On the fibration method for rational points