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Division points on semi-abelian varieties. (English) Zbl 0848.14022
Let $$A$$ be a semi-Abelian variety, defined over $$\mathbb{C}$$, and let $$\Gamma \subset A (\mathbb{C})$$ be any finitely generated group. The torsion hull $$\overline \Gamma$$ is defined by $$\overline \Gamma = \{x \in A (\mathbb{C}) \mid nx \in \Gamma$$, some $$n \in\mathbb{N}\}$$.
On the basis of conjectures of Mordell and Manin-Mumford, S. Lang made the following conjecture: Let $$X \subset A$$ be any closed integral subvariety such that $$X \cup \overline \Gamma$$ is Zariski dense in $$X$$. Then $$X$$ is the translate of a semi-Abelian variety by a point of $$\overline \Gamma$$. – A weaker case of the conjecture is that if $$X \subset A$$ is any closed subvariety, which does not contain any non null translated semi-Abelian varieties, then $$X \cap \overline \Gamma$$ is finite.
As the author points out, those conjectures may be thought of as in two parts: the ‘Mordellic part’, where $$\overline \Gamma$$ is replaced by $$\Gamma$$; the ‘torsion part’, when $$\Gamma = 0$$ and $$\overline \Gamma$$ is the torsion subgroup of $$A (\mathbb{C})$$. P. Vojta [“Integral points on subvarieties of semi-abelian varieties”, Invent. Math. (to appear)] generalized the work of G. Faltings [Ann. Math., II. Ser. 133, No. 3, 549-576 (1991; Zbl 0734.14007)] to prove the Mordellic part of the principal conjecture. The author uses the work of Vojta and M. Hindry [Invent. Math. 94, No. 3, 575-603 (1988; Zbl 0638.14026)] to prove the main conjecture.

MSC:
 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 14G05 Rational points 14K15 Arithmetic ground fields for abelian varieties
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References:
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