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Integral division points on curves
Part of:
Arithmetic algebraic geometry
Arithmetic problems. Diophantine geometry
Arithmetic and non-Archimedean dynamical systems
Published online by Cambridge University Press: 09 September 2013
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.
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