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Integral division points on curves. (English) Zbl 1292.11072
Let $$k$$ be a number field and $$S$$ a set of primes containing all the infinite ones. Let $$A/k$$ be a semi-abelian variety, $$\Gamma_0$$ a finitely generated subgroup of $$A(\overline k)$$ and $$\Gamma \subseteq A(\overline k)$$ be the division group attached to $$\Gamma_0$$, i.e the set of points $$P\in A(\overline k)$$ such that there exists a integer $$n$$ such that $$nP\in \Gamma_0$$.
If $$X/k$$ is any variety and $$\overline X$$ its completion, define $$\partial X:=\overline X - X$$. Let $$T$$ be any subset of $$\overline X$$, and let $$\overline T$$ be its Zariski closure in $$\overline X$$. Any $$P\in X(\overline k)$$ is said to be $$S$$-integral relative to $$T$$ if it is $$(\overline T \cup \partial X, S)$$-integral on $$\overline X$$.
The authors pose the following conjecture:
Conjecture Let $$k$$ and $$S$$ be as above and let $$A/k$$ be a semi-abelian variety and $$\Gamma$$ a division group in $$A(\overline k)$$. Suppose that $$D$$ is a non-zero effective divisor on $$A$$ which is not the translate of any torsion divisor by any point of $$\Gamma$$. Then the set $\{P\in \Gamma: P \text{ is $$S$$-integral relative to }D\}$ is not Zariski dense in $$A$$.
The authors then prove the conjecture for $$1$$-dimensional semi-abelian varieties, i.e. for elliptic curves and $$1$$-dimensional tori.

##### MSC:
 11G35 Varieties over global fields 11G05 Elliptic curves over global fields 11G50 Heights 14G05 Rational points 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 37P35 Arithmetic properties of periodic points
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