A finiteness property of torsion points

Let k be a number field, let E/k be an elliptic curve, and let S be a finite set of places of k contianing the archimedean places. Let F be an algebraic closure of k. We prove that if a point P in E(F) is nontorsion, then there are only finitely many torsion points x in E(F) which are S-integral with respect to P. We also prove an analogue of this for the multiplicative group, and formulate conjectural generalizations for abelian varieties and dynamical systems.

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