Algebra & Number Theory
- Algebra Number Theory
- Volume 11, Number 8 (2017), 1953-1965.
A Mordell–Weil theorem for cubic hypersurfaces of high dimension
Let be a smooth cubic hypersurface of dimension . 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 , Manin (1968) asked if there exists a finite set from which all other rational points can be thus obtained. We give an affirmative answer for , showing in fact that we can take the generating set 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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14G05: Rational points
Secondary: 11G35: Varieties over global fields [See also 14G25]
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. https://projecteuclid.org/euclid.ant/1510842829