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

Abstract

Let E be an elliptic curve. Consider an irreducible algebraic curve C embedded in E g . The curve is transverse if it is not contained in any translate of a proper algebraic subgroup of E g . Furthermore C is weak-transverse if it is not contained in any proper algebraic subgroup. Suppose that both E and C are defined over the algebraic numbers.We prove that the algebraic points of a transverse curve C which 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 close is defined using a height function. If Γ is trivial, it is sufficient to suppose that C is weak-transverse.Then, we introduce a method to determine the finiteness of these sets. From a conjectural lower bound for the normalised height of a transverse curve C, we deduce that the above sets are finite. At present, such a lower bound exists for g ≤ 3.Our results are optimal, for what concerns the codimension of the algebraic subgroups.S r+1 (V, F ) ⊂ S r (V, F ).

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