The optimality of the Bounded Height Conjecture
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)
@article{JTNB_2009__21_3_771_0,
     author = {Viada, Evelina},
     title = {The optimality of the {Bounded} {Height} {Conjecture}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {771--786},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.702},
     zbl = {1203.11048},
     mrnumber = {2605547},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.702/}
}
TY  - JOUR
AU  - Viada, Evelina
TI  - The optimality of the Bounded Height Conjecture
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 771
EP  - 786
VL  - 21
IS  - 3
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.702/
DO  - 10.5802/jtnb.702
LA  - en
ID  - JTNB_2009__21_3_771_0
ER  - 
%0 Journal Article
%A Viada, Evelina
%T The optimality of the Bounded Height Conjecture
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 771-786
%V 21
%N 3
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.702/
%R 10.5802/jtnb.702
%G en
%F JTNB_2009__21_3_771_0
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

Cité par Sources :