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**A finiteness property of torsion points.**
*(English)*
Zbl 1182.11030

Let \(k\) be a number field with algebraic closure \(\bar{k}\) and ring of integers \(\mathcal{O}_k\). Let \(S\) be a finite set of places of \(k\) containing the archimedean places.

Let \(\alpha,\beta\in\mathbb{P}^1(\bar{k})\) be points, then we say that \(\beta\) is \(S\)-integral relative to \(\alpha\) if the Zariski-closures of \(\alpha\) and \(\beta\) in \(\mathbb{P}^1_{\mathcal{O}_k}\) do not meet outside \(S\).

Similarly, if \(E/k\) is an elliptic curve with integral model \(\mathcal{E}/\mathrm{Spec}(\mathcal{O}_k)\), then a point \(\beta\in E(\bar{k})\) is \(S\)-integral relative to \(\alpha\in E(\bar{k})\) if the Zariski-closures of \(\alpha\) and \(\beta\) in \(\mathcal{E}\) do not meet outside fibres above \(S\).

The goal of the present paper is to prove the following results:

1) Let \(\alpha\in\mathbb{P}^1(\bar{k})\smallsetminus\{0,\infty\}\) be not a root of unity. Then there are only finitely many roots of unity in \(\bar{k}\) that are \(S\)-integral relative to \(\alpha\).

2) Let \(\alpha\in E(\bar{k})\) be a non-torsion point. Then \(E(\bar{k})\) contains only finitely many torsion points which are \(S\)-integral relative to \(\alpha\).

The main ingredients for the proofs of both results (which are structured similarly) are linear forms in logarithms (Baker’s theorem for \(\mathbb{G}_a\), and David/Hirata-Kohno’s theorem for elliptic curves, respectively), some properties of local height functions, and a strong form of equidistribution for torsion points at every place \(v\). The authors devote most of their energy to proving these strong equidistribution results, which are interesting in their own right.

The authors also place their results in the context of more general conjectures on dynamical systems and abelian varieties, and also show why their results cannot be strengthened in a naive way.

Let \(\alpha,\beta\in\mathbb{P}^1(\bar{k})\) be points, then we say that \(\beta\) is \(S\)-integral relative to \(\alpha\) if the Zariski-closures of \(\alpha\) and \(\beta\) in \(\mathbb{P}^1_{\mathcal{O}_k}\) do not meet outside \(S\).

Similarly, if \(E/k\) is an elliptic curve with integral model \(\mathcal{E}/\mathrm{Spec}(\mathcal{O}_k)\), then a point \(\beta\in E(\bar{k})\) is \(S\)-integral relative to \(\alpha\in E(\bar{k})\) if the Zariski-closures of \(\alpha\) and \(\beta\) in \(\mathcal{E}\) do not meet outside fibres above \(S\).

The goal of the present paper is to prove the following results:

1) Let \(\alpha\in\mathbb{P}^1(\bar{k})\smallsetminus\{0,\infty\}\) be not a root of unity. Then there are only finitely many roots of unity in \(\bar{k}\) that are \(S\)-integral relative to \(\alpha\).

2) Let \(\alpha\in E(\bar{k})\) be a non-torsion point. Then \(E(\bar{k})\) contains only finitely many torsion points which are \(S\)-integral relative to \(\alpha\).

The main ingredients for the proofs of both results (which are structured similarly) are linear forms in logarithms (Baker’s theorem for \(\mathbb{G}_a\), and David/Hirata-Kohno’s theorem for elliptic curves, respectively), some properties of local height functions, and a strong form of equidistribution for torsion points at every place \(v\). The authors devote most of their energy to proving these strong equidistribution results, which are interesting in their own right.

The authors also place their results in the context of more general conjectures on dynamical systems and abelian varieties, and also show why their results cannot be strengthened in a naive way.

Reviewer: Florian Breuer (Stellenbosch)