A geometric formulation of Siegel's diophantine theorem

TL;DR

This paper presents an algebro-geometric reformulation of Siegel's theorem, extending finiteness results for intersections of finitely generated groups with affine curves to a broader algebraic setting.

Contribution

It introduces a new geometric formulation of Siegel's theorem using an improved version of Roth's theorem over finitely generated fields, generalizing previous results.

Findings

Finitely generated subgroups intersect affine curves in finitely many points

Extension of finiteness results beyond number fields

Connection between Diophantine approximation and algebraic geometry

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.