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 suficient 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

Primary: 11G05

Secondary: 11D45, 11G50, 14K12

Authors