Algebra & Number Theory

A Mordell–Weil theorem for cubic hypersurfaces of high dimension

Stefanos Papanikolopoulos and Samir Siksek

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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.

Article information

Algebra Number Theory, Volume 11, Number 8 (2017), 1953-1965.

Received: 10 October 2016
Revised: 13 July 2017
Accepted: 11 August 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G05: Rational points
Secondary: 11G35: Varieties over global fields [See also 14G25]

cubic hypersurfaces rational points Mordell–Weil problem


Papanikolopoulos, Stefanos; Siksek, Samir. A Mordell–Weil theorem for cubic hypersurfaces of high dimension. Algebra Number Theory 11 (2017), no. 8, 1953--1965. doi:10.2140/ant.2017.11.1953.

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