  We prove a special case of the following conjecture of Zilber-Pink
generalising the Manin-Mumford conjecture : let $X$ be a curve inside an
Abelian variety $A$ over $\bar{\Q}$, provided $X$ is not contained in a torsion
subvariety, the intersection of $X$ with the union of all subgroup schemes of
codimension at least 2 is finite ; we settle the case where $A$ is a power of a
simple Abelian variety of C.M. type. This generalises the previous known
result, due to Viada and R\'emond-Viada (who was able to prove the conjecture
for power of an elliptic curve with complex multiplication). The proof is based
on the strategy of R\'emond (following Bombieri, Masser and Zannier) with two
new ingredients, one of them, being at the heart of this article : it is a
lower bound for the N\'eron-Tate height of points on Abelian varieties $A/K$ of
C.M. type in the spirit of Lehmer's problem. This lower bound is an analog of
the similar result of Amoroso and David \cite{ad2003} on $\G_m^n$ and is a
generalisation of the theorem of David and Hindry \cite{davidhindry} on the
abelian Lehmer's problem. The proof is an adaptation of \cite{davidhindry}
using in our abelian case the new ideas introduced in \cite{ad2003}.
Furthermore, as in \cite{ad2003} and adapting in the abelian case their proof,
we give another application of our result : a 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.
