Borne sur la torsion dans les vari\'et\'es ab\'eliennes de type C.M

Let A be an abelian variety of dimension g defined over a number field K. We study the size of the torsion group A(F)_{tors} where F/K is a finite extension and more precisely we study the possible exponent \gamma in the inequality Card(A(F)_{tors})<< [F:K]^{\gamma} when F is any extension of K. In the C.M. case we give an exact formula for the best possible exponent in terms of the characters of the Mumford-Tate group--a torus in this case--and discuss briefly the general case. Finally we give applications of this result in direction of a conjecture of R\'emond generalising the Manin-Mumford conjecture.