51 pagesIn this article we give a lower bound for the Néron-Tate height of points on Abelian varieties A/K of C.M. type in the spirit of Lehmer's problem. Our result is a generalisation of the theorem of David and Hindry on the abelian Lehmer's problem. Furthermore we give two applications of our result : the first is a new lower bound for the absolute minimum of a subvariety V of A. Although lower bounds for this minimum were already known (decreasing multi-exponential function of the degree for Bombieri-Zannier), our methods enable us to prove, up to an epsilon the optimal result that can be conjectured. The second application is a theorem in the direction of a conjecture of Rémond generalising the Manin-Mumford conjecture : we prove Rémond's conjecture for all power of one simple Abelian variety of C.M. type of dimension g>0. This generalises the previous known result, due to Viada (who was able to prove Rémond's conjecture for power of one elliptic curve with complex multiplication) concerning this problem
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