A finiteness property of torsion points

Abstract

Let $k$ be a number field, and let $G$ be either the multiplicative group ${\mathbb{G}}_m∕k$ or an elliptic curve $E∕k$. Let $S$ be a finite set of places of $k$ containing the archimedean places. We prove that if $\alpha \in G\left(\overline{k}\right)$ is nontorsion, then there are only finitely many torsion points $\xi \in G{\left(\overline{k}\right)}_{tors}$ that are $S$-integral with respect to $\alpha$. We also formulate conjectural generalizations for dynamical systems and for abelian varieties.