Vol. 2, No. 3, 2008

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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
Abstract

Let E be an elliptic curve. An irreducible algebraic curve C embedded in Eg is called weak-transverse if it is not contained in any proper algebraic subgroup of Eg, 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 Eg of codimension 2 translated by points in a subgroup Γ of Eg 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 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