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 ).

Abstract. Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V . Especially, we determine whether such a set is or not dense in V . We then prove that these sets can always be written as the intersection of V with a finite union of translates of tori of which we control the sum of the degrees.As a consequence, we prove a conjecture by the first author and David up to a logarithmic factor.

Let p > 3 be a prime number and let n be a positive integer. We prove that the local-global principle for divisibility by p n holds for elliptic curves defined over the rationals. For this, we refine our previous criterion for the validity of the principle. We also give an example that shows that the assumptions of our criterion are necessary.

Abstract. The Torsion Anomalous Conjecture states that an irreducible variety V embedded in a semi-abelian variety contains only finitely many maximal V -torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a produc of elliptic curves. Our main result provides a totally explicit bound for the Néron-Tate height of all maximal V -torsion anomalous points of relative codimension one, in the non CM case, and an analogous effective result in the CM case. As an application, we obtain the finiteness of such points. In addition, we deduce some new explicit results in the context of the effective Mordell-Lang Conjecture; in particular we bound the Néron-Tate height of the rational points of an explicit family of curves of increasing genus.