zbMATH — the first resource for mathematics

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.

14K20 Analytic theory of abelian varieties; abelian integrals and differentials
14G05 Rational points
14K15 Arithmetic ground fields for abelian varieties
Full Text: DOI EuDML
[1] Abramovich, D.: Subvarieties of Semi-Abelian Varieties. Comp. Math.90, 37-57 (1994) · Zbl 0814.14041
[2] Faltings, G.: Diophantine Approximation on Abelian Varieties. Ann. of Math.131, 549-576 (1991) · Zbl 0734.14007 · doi:10.2307/2944319
[3] Faltings, G.: Finiteness Theorems for Abelian Varieties over Number Fields. In: Cornell, G., Silverman, J. (eds.): Arithmetic Geometry. Berlin Heidelberg New York: Springer 1986 · Zbl 0602.14044
[4] Hindry, M.: Antour d’une Conjecture de Serge Lang. Invent. Math.94, 575-603 (1988) · Zbl 0638.14026 · doi:10.1007/BF01394276
[5] Lang, S.: Division Points on Curves. Ann. Mat. Pura Appl.LXX, 229-234 (1965) · Zbl 0151.27401
[6] Lang, S.: Fundamentals of Diophantine Geometry. Berlin Heidelberg New York: Springer 1983 · Zbl 0528.14013
[7] Lang, S.: Volume for the encyclopedia of Mathematics on Diophantine Geometry. Berlin Heidelberg New York: Springer 1992
[8] Laurent, M.: Equations Diophantinnes Exponentielles. Invent. Math.78, 299-327 (1984) · Zbl 0554.10009 · doi:10.1007/BF01388597
[9] Raynaud, M.: Sous-Variétés d’une variété abélienne et points de torsion. In: Coates, J., Helgason, S. (eds.): Arithmetic & Geometry, Papers dedicated to I.R. Shafarevich on the Occasion of his Sixtieth Birthday, Volume 1, Birkhäuser 1983
[10] Serre, J.P.: Quelques propriétés des groupes algébriques commutatifs. Appendix in Astérisque69-70, 191-202 (1979)
[11] Serre, JP.: Résumé des Cours au Collège de France (1985-86). Ann. Collège de France, 95-99 (1986)
[12] Vojta, P.: Integral Points on Subvarieties of Semi-Abelian Varieties. Invent. Math. (to appear) · Zbl 1011.11040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.