The Hardy--Littlewood conjecture and rational points

TL;DR

This paper demonstrates that under certain conditions, the Hardy-Littlewood conjecture can be used to obtain unconditional results on rational points, replacing the need for Schinzel's Hypothesis (H) in specific cases.

Contribution

It shows that the methods relying on Schinzel's Hypothesis (H) can be replaced with recent results on the generalized Hardy-Littlewood conjecture for certain rational varieties.

Findings

Unconditional results on rational points are obtained for specific varieties.

The approach applies when degenerate fibers are defined over Q.

Recent advances in the Hardy-Littlewood conjecture enable these results.

Abstract

Schinzel's Hypothesis (H) was used by Colliot-Th\'el\`ene 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.