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arXiv · 1605.02787

A Mordell-Weil theorem for cubic hypersurfaces of high dimension

Stefanos Papanikolopoulos·Samir Siksek
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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.

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Stefanos Papanikolopoulos, Samir Siksek. 2016-05-09. A Mordell-Weil theorem for cubic hypersurfaces of high dimension. https://doi.org/10.2140/ant.2017.11.1953

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