The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve

Viada, Evelina

doi:10.2140/ant.2008.2.249

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The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve

Evelina Viada

Vol. 2 (2008), No. 3, 249–298

DOI: 10.2140/ant.2008.2.249

Abstract

Let $E$ be an elliptic curve. An irreducible algebraic curve $C$ embedded in $E^{g}$ is called weak-transverse if it is not contained in any proper algebraic subgroup of $E^{g}$ , and transverse if it is not contained in any translate of such a subgroup.

Suppose $E$ and $C$ are defined over the algebraic numbers. First we prove that the algebraic points of a transverse curve $C$ that are close to the union of all algebraic subgroups of $E^{g}$ of codimension $2$ translated by points in a subgroup $Γ$ of $E^{g}$ of finite rank are a set of bounded height. The notion of closeness is defined using a height function. If $Γ$ is trivial, it is sufficient to suppose that $C$ is weak-transverse.

The core of the article is the introduction of a method to determine the finiteness of these sets. From a conjectural lower bound for the normalized height of a transverse curve $C$ , we deduce that the sets above are finite. Such a lower bound exists for $g \leq 3$ .

Concerning the codimension of the algebraic subgroups, our results are best possible.

Keywords

heights, diophantine approximation, elliptic curves, counting algebraic points

Mathematical Subject Classification 2000

Primary: 11G05

Secondary: 11D45, 11G50, 14K12

Milestones

Received: 12 April 2007

Revised: 2 April 2008

Accepted: 4 April 2008

Published: 1 May 2008

Authors

Evelina Viada
Université de Fribourg Suisse, Pérolles Département de Mathématiques 23 Chemin du Musée CH-1700 Fribourg Suisse

Vol. 2, No. 3, 2008