A Mordell-Weil theorem for cubic hypersurfaces of high dimension

TL;DR

This paper proves that for high-dimensional smooth cubic hypersurfaces over rationals, all rational points can be generated from a single point using secant and tangent methods, extending Mordell-Weil type results.

Contribution

It establishes a Mordell-Weil type theorem for cubic hypersurfaces of dimension at least 48, showing a single rational point suffices to generate all others.

Findings

All rational points can be generated from one point for n ≥ 48.

Uses a combination of algebraic geometry, differential geometry, and number theory.

Proves a finite generation property similar to Mordell-Weil theorem for higher dimensions.

Abstract

Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. 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 $n \ge 48$, 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.