Vol. 11, No. 8, 2017

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A Mordell–Weil theorem for cubic hypersurfaces of high dimension

Stefanos Papanikolopoulos and Samir Siksek

Vol. 11 (2017), No. 8, 1953–1965
Abstract

Let X be a smooth cubic hypersurface of dimension n 1. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell–Weil theorem for n = 1, Manin (1968) asked if there exists a finite set S from which all other rational points can be thus obtained. We give an affirmative answer for n 48, showing in fact that we can take the generating set S to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on the existence of rational points on the intersection of a quadric and cubic in large dimension, and some elementary ideas from differential geometry, algebraic geometry and numerical analysis.

Keywords
cubic hypersurfaces, rational points, Mordell–Weil problem
Mathematical Subject Classification 2010
Primary: 14G05
Secondary: 11G35
Milestones
Received: 10 October 2016
Revised: 13 July 2017
Accepted: 11 August 2017
Published: 15 October 2017
Authors
Stefanos Papanikolopoulos
Mathematics Institute
University of Warwick
Coventry
United Kingdom
Samir Siksek
Mathematics Institute
University of Warwick
Coventry
United Kingdom