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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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