Vol. 2, No. 2, 2008

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A finiteness property of torsion points

Matthew Baker, Su-ion Ih and Robert Rumely

Vol. 2 (2008), No. 2, 217–248
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.

Keywords
elliptic curve, equidistribution, canonical height, torsion point, integral point
Mathematical Subject Classification 2000
Primary: 11G05
Secondary: 11J71, 11J86, 37F10, 11G50