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Integral division points on curves

Part of: Arithmetic problems. Diophantine geometry Arithmetic and non-Archimedean dynamical systems Arithmetic algebraic geometry

Published online by Cambridge University Press:  09 September 2013

David Grant  and
Su-Ion Ih
David Grant
Affiliation:
Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0395, USA email [email protected]
Su-Ion Ih
Affiliation:
Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0395, USA email [email protected]
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Abstract

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Let $k$ be a number field with algebraic closure $ \overline{k} $, and let $S$ be a finite set of primes of $k$ containing all the infinite ones. Let $E/ k$ be an elliptic curve, ${\mit{\Gamma} }_{0} $ be a finitely generated subgroup of $E( \overline{k} )$, and $\mit{\Gamma} \subseteq E( \overline{k} )$ the division group attached to ${\mit{\Gamma} }_{0} $. Fix an effective divisor $D$ of $E$ with support containing either: (i) at least two points whose difference is not torsion; or (ii) at least one point not in $\mit{\Gamma} $. We prove that the set of ‘integral division points on $E( \overline{k} )$’, i.e., the set of points of $\mit{\Gamma} $ which are $S$-integral on $E$ relative to $D, $ is finite. We also prove the ${ \mathbb{G} }_{\mathrm{m} } $-analogue of this theorem, thereby establishing the 1-dimensional case of a general conjecture we pose on integral division points on semi-abelian varieties.

Keywords

division groupdivision pointintegral pointprimitive divisorSchinzel’s theoremSiegel’s theorem

MSC classification

Secondary: 11G05: Elliptic curves over global fields 11G35: Varieties over global fields 11G50: Heights 14G05: Rational points 37P05: Polynomial and rational maps 37P35: Arithmetic properties of periodic points
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 12 , December 2013 , pp. 2011 - 2035
Copyright
© The Author(s) 2013 

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