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Evelina Viada
The optimality of the Bounded Height Conjecture
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 771-786
Article: subscription required
Class. Math.: 11G50, 14H52, 14K12
Keywords: Height, Elliptic curves, Subvarieties
[1] E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups. Int. Math. Res. Not. 20 (1999), 1119–1140. MR 1728021 | Zbl 0938.11031
[2] E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties - Structure Theorem and applications. Int. Math. Res. Not. 19 (2007), 33 pages. MR 2359537 | Zbl 1145.11049
[3] P. Habegger, Bounded height for subvarieties in abelian varieties. Invent. math. 176 (2009), 405–447. Zbl 1176.14008
[4] G. Rémond, Intersection de sous-groupes et de sous-variétés II. J. Inst. Math. Jussieu 6 (2007), 317–348. MR 2311666 | Zbl 1170.11014
[5] G. Rémond, Intersection de sous-groups et de sous-variétés III. To appear in Com. Mat. Helv. MR 2534482 | Zbl pre05609487
[6] G. Rémond and E. Viada, Problème de Mordell-Lang modulo certaines sous-variétés abéliennes. Int. Math. Res. Not. 35 (2003), 1915–1931. MR 1995142 | Zbl 1072.11038
[7] E. Viada, The intersection of a curve with algebraic subgroups in a product of elliptic curves. Ann. Scuola Norm. Sup. Pisa cl. Sci. 5 vol. II (2003), 47–75.
Numdam | MR 1990974 | Zbl 1170.11314
[8] E. Viada, The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve, Algebra and Number Theory 3 vol. 2 (2008), 248–298. MR 2407116 | Zbl 1168.11024
[9] E. Viada, Non-dense subsets of varieties in a power of an elliptic curve. Int. Math. Res. Not. 7 (2009), 1214–1246. MR 2495303 | Zbl 1168.14030
Evelina Viada
The optimality of the Bounded Height Conjecture
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 771-786
Article: subscription required
Class. Math.: 11G50, 14H52, 14K12
Keywords: Height, Elliptic curves, Subvarieties
Résumé - Abstract
In this article we show that the Bounded Height Conjecture is optimal in the sense that, if $V$ is an irreducible subvariety with empty deprived set in a power of an elliptic curve, then every open subset of $V$ does not have bounded height. The Bounded Height Conjecture is known to hold. We also present some examples and remarks.
Bibliography
[2] E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties - Structure Theorem and applications. Int. Math. Res. Not. 19 (2007), 33 pages. MR 2359537 | Zbl 1145.11049
[3] P. Habegger, Bounded height for subvarieties in abelian varieties. Invent. math. 176 (2009), 405–447. Zbl 1176.14008
[4] G. Rémond, Intersection de sous-groupes et de sous-variétés II. J. Inst. Math. Jussieu 6 (2007), 317–348. MR 2311666 | Zbl 1170.11014
[5] G. Rémond, Intersection de sous-groups et de sous-variétés III. To appear in Com. Mat. Helv. MR 2534482 | Zbl pre05609487
[6] G. Rémond and E. Viada, Problème de Mordell-Lang modulo certaines sous-variétés abéliennes. Int. Math. Res. Not. 35 (2003), 1915–1931. MR 1995142 | Zbl 1072.11038
[7] E. Viada, The intersection of a curve with algebraic subgroups in a product of elliptic curves. Ann. Scuola Norm. Sup. Pisa cl. Sci. 5 vol. II (2003), 47–75.
Numdam | MR 1990974 | Zbl 1170.11314
[8] E. Viada, The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve, Algebra and Number Theory 3 vol. 2 (2008), 248–298. MR 2407116 | Zbl 1168.11024
[9] E. Viada, Non-dense subsets of varieties in a power of an elliptic curve. Int. Math. Res. Not. 7 (2009), 1214–1246. MR 2495303 | Zbl 1168.14030
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