Torsion for abelian varieties of type III

Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$ of $L$ over $K$. Under the following three conditions, we compute the optimal exponent for this bound, in terms of the dimension of abelian subvarieties and their endomorphism rings. The three hypothesis are the following: $(1)$ $A$ is geometrically isogenous to a product of simple abelian varieties of type I, II or III, according to the Albert classification; $(2)$ $A$ is of "Lefschetz type", that is, the Mumford-Tate group is the group of symplectic or orthogonal similitudes which commute with the endomorphism ring; $(3)$ $A$ satisfies the Mumford-Tate conjecture. This result is unconditional for a product of simple abelian varieties of type I, II or III with specific relative dimensions. Further, building on work of Serre, Pink, Banaszak, Gajda and Kraso\'n, we also prove the Mumford-Tate conjecture for a few new cases of abelian varieties of Lefschetz type.