A Mordell–Weil theorem for cubic hypersurfaces of high dimension

Papanikolopoulos, Stefanos; Siksek, Samir

doi:10.2140/ant.2017.11.1953

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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

DOI: 10.2140/ant.2017.11.1953

Abstract

Let $X ∕ ℚ$ be a smooth cubic hypersurface of dimension $n \geq 1$ . 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 $n \geq 48$ , 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

Vol. 11, No. 8, 2017