Deux remarques sur le probleme de Lehmer sur les varietes abeliennes

Abstract

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 N\'eron-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 \cite{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 N\'eron-Tate height of $n$ End$(A)$-linearly independant non-torsion points of a C.M. type abelian variety.