Deux remarques sur le probleme de Lehmer sur les varietes abeliennes

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Let A/K be an abelian variety over a number field K. We prove in this article that a good lower bound (in terms of the degree [K(P):K]) for the Néron-Tate height of the points P of infinite order modulo every strict abelian subvarieties of A implies a good lower bound for the height of all the non-torsion points of A. In particular when A is of C.M. type, a theorem of David and Hindry enables us to deduce, up to ``log'' factors, an optimal lower bound for the height of the non-torsion points of A. In the C.M. type case, this improves the previous result of Masser lettre. Using the same theorem of David and Hindry we prove in the second part an optimal lower bound, up to ``log'' factors, for the product of the Néron-Tate height of n End(A)-linearly independant non-torsion points of a C.M. type abelian variety.