## Algebra & Number Theory

### A Mordell–Weil theorem for cubic hypersurfaces of high dimension

#### Abstract

Let $X∕ℚ$ 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

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

Dates
Revised: 13 July 2017
Accepted: 11 August 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842829

Digital Object Identifier
doi:10.2140/ant.2017.11.1953

Mathematical Reviews number (MathSciNet)
MR3720936

Zentralblatt MATH identifier
06806366

Subjects
Primary: 14G05: Rational points

#### Citation

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

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