Vol. 2, No. 3, 2008

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 10, 2053–2310
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
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

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.

heights, diophantine approximation, elliptic curves, counting algebraic points
Mathematical Subject Classification 2000
Primary: 11G05
Secondary: 11D45, 11G50, 14K12
Received: 12 April 2007
Revised: 2 April 2008
Accepted: 4 April 2008
Published: 1 May 2008
Evelina Viada
Université de Fribourg Suisse, Pérolles
Département de Mathématiques
23 Chemin du Musée
CH-1700 Fribourg