Topics in Diophantine Equations

Swinnerton-Dyer, Sir Peter

doi:10.1007/978-3-642-15945-9_2

Cited by 3 publications

(1 citation statement)

References 34 publications

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“…The method of [CSan82] was extended to number fields and generalised in various directions by Serre (see [Serre], Ch. II, Annexe), Swinnerton-Dyer (see [Sw94], [Sw09] and references in that paper), Colliot-Thélène and others [CSw94,CSkSw98a,W07,Wei12]. In this note we show that when the ground field is Q and the degenerate geometric fibres of the pencil are all defined over Q, one can apply the methods of the aforementioned papers to obtain unconditional results by using the theorem of Green, Tao and Ziegler in place of Schinzel's Hypothesis (H).…”

Section: Introductionmentioning

confidence: 99%

The Hardy–Littlewood conjecture and rational points

Harpaz¹,

Skorobogatov²,

Wittenberg³

2014

Compositio Math.

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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 Q and the degenerate geometric fibres of the pencil are all defined over 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.

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Section: Introductionmentioning

confidence: 99%

The Hardy–Littlewood conjecture and rational points

Harpaz¹,

Skorobogatov²,

Wittenberg³

2014

Compositio Math.

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On the fibration method for zero-cycles and rational points

Harpaz¹,

Wittenberg²

2016

Ann. Math.

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Abstract. Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Thélène, Sansuc, Kato and Saito in the 1980's. We prove that these conjectures are compatible with fibrations, for fibrations into rationally connected varieties over a curve. In particular, they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers. We prove the analogous result for rational points, conditionally on a conjecture on locally split values of polynomials which a recent work of Matthiesen establishes in the case of linear polynomials over the rationals.

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Application of the explicit abc-conjecture to two Diophantine equations

Saradha

2012

Acta Arith.

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Topics in Diophantine Equations

Cited by 3 publications

References 34 publications

The Hardy–Littlewood conjecture and rational points

The Hardy–Littlewood conjecture and rational points

On the fibration method for zero-cycles and rational points

Application of the explicit abc-conjecture to two Diophantine equations

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