A uniform bound for the p-primary torsion in abelian threefolds of Picard type

In 1969 Manin proved a uniform version of Serre's celebrated result on the openness of the Galois image in the automorphisms of the $\ell$-adic Tate module of any non-CM elliptic curve over a given number field. Recently, in a series of papers Cadoret and Tamagawa established a definitive uniform open image result for the $\ell$-adic representations attached to $1$-dimensional abelian families. In this lecture, based on joint work with M. Dimitrov, we provide the first evidence in higher dimension, for families of abelian 3-folds parametrized by certain Picard modular surfaces.