Minoration effective de la hauteur des points d'une courbe de $G\mathrm{\_}{m}^2$ d\'{e}finie sur $Q$

We are concerned here with Lehmer’s problem in dimension 2 ; we give a lower bound for the height of a non-torsion point of $G\mathrm{\_}{m}^2$ on a non-torsion curve defined over $Q$, depending on the degree of the curve only. We have first been inspired by \cite{Am-Da3}; we develop a new approach, inherent in the dimension two (or more precisely the codimension two), and then obtain a better result where the error’s term is improved significantly, moreover we give an explicit expression for the constant.