A finiteness property of torsion points

Baker, Matthew; Ih, Su-ion; Rumely, Robert

doi:10.2140/ant.2008.2.217

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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

DOI: 10.2140/ant.2008.2.217

Abstract

Let $k$ be a number field, and let $G$ be either the multiplicative group $G_{m} ∕ k$ or an elliptic curve $E ∕ k$ . Let $S$ be a finite set of places of $k$ containing the archimedean places. We prove that if $α \in G (\bar{k})$ is nontorsion, then there are only finitely many torsion points $ξ \in G {(\bar{k})}_{tors}$ that are $S$ -integral with respect to $α$ . 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

Milestones

Received: 29 October 2007

Revised: 11 January 2008

Accepted: 11 January 2008

Published: 15 March 2008

Authors

Matthew Baker
School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332-0160 United States

Su-ion Ih
Department of Mathematics University of Colorado at Boulder Campus Box 395 Boulder, CO 80309-0395 United States

Robert Rumely
Department of Mathematics University of Georgia Athens, Georgia 30602-0002 United States

Vol. 2, No. 2, 2008