  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.
