The Hilbert Property for integral points of affine smooth cubic surfaces

Abstract

In this paper we prove that the set of $S$-integral points of the smooth cubic surfaces in ${\mathbb{A}}^3$ over a number field $k$ is not thin, for suitable $k$ and $S$. As a corollary, we obtain results on the complement in ${\mathbb{P}}^2$ of a smooth cubic curve, improving on Beukers' proof that the $S$-integral points are Zariski dense, for suitable $S$ and $k$. With our method we reprove Zariski density, but our result is more powerful since it is a stronger form of Zariski density. We moreover prove that the rational integer points on the Fermat cubic surface $x^3+y^3+z^3=1$ form a non-thin set and we link our methods to previous results of Lehmer, Miller-Woollett and Mordell.

18 pages

Cited by in corpus (3)

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