A geometric formulation of Siegel's diophantine theorem

Abstract

In this paper, we introduce an algebro-geometric formulation for Siegel's theorem using an improvement of Lang's version of Roth's theorem over finitely generated fields of characteristic zero. In fact, we prove that, for an affine open curve in an irreducible smooth curve of genus at least one, any finitely generated subgroup of the additive group of the affine ambient space intersects the open curve in only finitely many points. This was proved only for finitely generated subgroups defined over a localization of the ring of integers of a number field by Mahler and others.