# The optimality of the Bounded Height Conjecture

Evelina Viada^{[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)

Journal de Théorie des Nombres de Bordeaux (2009)

- Volume: 21, Issue: 3, page 771-786
- ISSN: 1246-7405

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topViada, Evelina. "The optimality of the Bounded Height Conjecture." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 771-786. <http://eudml.org/doc/10912>.

@article{Viada2009,

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.},

affiliation = {Université de Fribourg Suisse, Pérolles Département de Mathématiques Chemin du Musée 23 CH-1700 Fribourg, Switzerland Supported by the SNF (Swiss National Science Foundation)},

author = {Viada, Evelina},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {Height; Elliptic curves; Subvarieties; height; elliptic curves; subvarieties},

language = {eng},

number = {3},

pages = {771-786},

publisher = {Université Bordeaux 1},

title = {The optimality of the Bounded Height Conjecture},

url = {http://eudml.org/doc/10912},

volume = {21},

year = {2009},

}

TY - JOUR

AU - Viada, Evelina

TI - The optimality of the Bounded Height Conjecture

JO - Journal de Théorie des Nombres de Bordeaux

PY - 2009

PB - Université Bordeaux 1

VL - 21

IS - 3

SP - 771

EP - 786

AB - 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.

LA - eng

KW - Height; Elliptic curves; Subvarieties; height; elliptic curves; subvarieties

UR - http://eudml.org/doc/10912

ER -

## References

top- E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups. Int. Math. Res. Not. 20 (1999), 1119–1140. Zbl0938.11031MR1728021
- E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties - Structure Theorem and applications. Int. Math. Res. Not. 19 (2007), 33 pages. Zbl1145.11049MR2359537
- P. Habegger, Bounded height for subvarieties in abelian varieties. Invent. math. 176 (2009), 405–447. Zbl1176.14008
- G. Rémond, Intersection de sous-groupes et de sous-variétés II. J. Inst. Math. Jussieu 6 (2007), 317–348. Zbl1170.11014MR2311666
- G. Rémond, Intersection de sous-groups et de sous-variétés III. To appear in Com. Mat. Helv. Zbl1227.11078MR2534482
- G. Rémond and E. Viada, Problème de Mordell-Lang modulo certaines sous-variétés abéliennes. Int. Math. Res. Not. 35 (2003), 1915–1931. Zbl1072.11038MR1995142
- 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. Zbl1170.11314MR1990974
- 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. Zbl1168.11024MR2407116
- E. Viada, Non-dense subsets of varieties in a power of an elliptic curve. Int. Math. Res. Not. 7 (2009), 1214–1246. Zbl1168.14030MR2495303

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