The optimality of the Bounded Height Conjecture
Viada, Evelina1
1 Université de Fribourg Suisse, Pérolles Département de Mathématiques Chemin du Musée 23 CH-1700 Fribourg, Switzerland Supported by the SNF (Swiss National Science Foundation)
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 771-786.

Nous démontrons que la “conjecture de hauteur bornée” est optimale dans le sens suivant. Soit V une variété irréductible dans une puissance d’une courbe elliptique. Si les sous-variétés anormales de V recouvrent tout V, alors chaque ouvert de V a une hauteur non bornée. Nous donnons aussi quelques exemples

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.

DOI : 10.5802/jtnb.702
Classification : 11G50, 14H52, 14K12
Mots-clés : Height, Elliptic curves, Subvarieties
Viada, Evelina 1

1 Université de Fribourg Suisse, Pérolles Département de Mathématiques Chemin du Musée 23 CH-1700 Fribourg, Switzerland Supported by the SNF (Swiss National Science Foundation) 
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Viada, Evelina. The optimality of the Bounded Height Conjecture. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 771-786. doi : 10.5802/jtnb.702. http://archive.numdam.org/articles/10.5802/jtnb.702/

[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 | Zbl

[2] E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties - Structure Theorem and applications. Int. Math. Res. Not. 19 (2007), 33 pages. | MR | Zbl

[3] P. Habegger, Bounded height for subvarieties in abelian varieties. Invent. math. 176 (2009), 405–447. | Zbl

[4] G. Rémond, Intersection de sous-groupes et de sous-variétés II. J. Inst. Math. Jussieu 6 (2007), 317–348. | MR | Zbl

[5] G. Rémond, Intersection de sous-groups et de sous-variétés III. To appear in Com. Mat. Helv. | MR

[6] G. Rémond and E. Viada, Problème de Mordell-Lang modulo certaines sous-variétés abéliennes. Int. Math. Res. Not. 35 (2003), 1915–1931. | MR | Zbl

[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 | Zbl

[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 | Zbl

[9] E. Viada, Non-dense subsets of varieties in a power of an elliptic curve. Int. Math. Res. Not. 7 (2009), 1214–1246. | MR | Zbl

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