A Mordell-Weil theorem for cubic hypersurfaces of high dimension

Samir Siksek (University of Warwick)

Wednesday 15th January, 2020 16:00-17:00 Maths 311B

Abstract

Let X be a smooth cubic hypersurface 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 in dimension 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 dimension at least 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. This is joint work with Stefanos Papanikolopoulos.

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