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

Let E/K be an elliptic curve with complex multiplication and let $K^{ab}$ be the Abelian closure of $K$. We prove in this article that there exists a constant $c(E\mathrm{/}K)$ such that : for all point $P\in E(\bar{K})-E_{tors}$, we have $$\widehat{h}(P)\ge \frac{c(E\mathrm{/}K)}{D}(\frac{\log \log 5D}{\log 2D}{)}^{13},$$ where $D=[K^{ab}(P):K^{ab}]$. This result extends to the case of elliptic curve s with complex multiplication the previous resultof Amoroso-Zannier \cite{AZ} on the analogous problem on the multiplicative group ${\mathbb{G}}_m$, and generalizes to the case of extensions of degree D the result of Baker \cite{baker} on the lower bound of the N'eron-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 \cite{viada}.