Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue.Using the elliptic Baker's method, we have recently obtained anew upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a  bound relying on the conjecture of Birch and Swinnerton-Dyer,involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.