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 $c(E/K)$ such that : for all point $P\in E(\bar{ K})-E_{ tors}$, we have \[\hat{ h}(P)\geq\frac{ c(E/K)}{ D}(\frac{ \log \log 5D}{ \log 2D})^{ 13},\] 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}.
Categories: math.NT math.AG