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

Let E be an elliptic curve. An irreducible algebraic curve C embedded in a power A of E is called weak-transverse if it is not contained in any proper algebraic subgroup of A, 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 G of A of finite rank are a set of bounded height. The notion of closeness is defined using a height function. If G 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 finite- ness 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.