  Let $A$ be an abelian variety defined over a number field $K$, the number of
torsion points rational over a finite extension $L$ is bounded polynomially in
terms of the degree $[L:K]$. We formulate a question suggesting the optimal
exponent for this bound in terms of the dimension of the Mumford-Tate groups of
the abelian subvarieties of $A$; we study the behaviour under product and then
give a positive answer to our question when $A$ is the product of elliptic
curves.
