Theoreme de Dobrowolski-Laurent pour les extensions abeliennes sur une courbe elliptique a multiplication complexe

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Let E/K be an elliptic curve with complex multiplication and let Kab be the Abelian closure of K. We prove in this article that there exists a constant c(E/K) such that: for all point P∈ E(K)-Etors, we have h(P)≥c(E/K)/D(5D/ 2D)¹³, where D=[Kab(P):Kab]. This result extends to the case of elliptic curve s with complex multiplication the previous resultof Amoroso-Zannier AZ on the analogous problem on the multiplicative group Gₘ, and generalizes to the case of extensions of degree D the result of Baker baker on the lower bound of the Néron-Tate height of the points defined over an Abelian extension of an elliptic curve with complex multiplication. This result also enables us to simplify the proof of a theorem of Viada viada.