Points de torsion sur les varietes abeliennes de type GSp

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 $[L:K]$. When $A$ is isogenous to a product of simple abelian varieties of $\GSp$ type, i.e. whose Mumford-Tate group is "generic" (isomorphic to the group of symplectic similitudes) and which satisfy the Mumford-Tate conjecture, we compute the optimal exponent for this bound in terms of the dimensions of the abelian subvarieties of $A$. The result is unconditional for a product of simple abelian varieties with endomorphism ring $\Z$ and dimension outside an explicit exceptional set $\mathcal{S}=\{4,10,16,32,...\}$. Furthermore, following a strategy of Serre, we also prove that if the Mumford-Tate conjecture is true for some abelian varieties of $\GSp$ type, it is then true for a product of such abelian varieties.