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Let $A$ be an abelian variety defined over a number field $K$, the number oftorsion points rational over a finite extension $L$ is bounded polynomially interms of the degree $[L:K]$. When $A$ is isogenous to a product of simpleabelian varieties of $\GSp$ type, i.e. whose Mumford-Tate group is "generic"(isomorphic to the group of symplectic similitudes) and which satisfy theMumford-Tate conjecture, we compute the optimal exponent for this bound interms of the dimensions of the abelian subvarieties of $A$. The result isunconditional 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 abelianvarieties of $\GSp$ type, it is then true for a product of such abelianvarieties.

Points de torsion sur les variétés abéliennes de type GSp