Uniform irreducibility of Galois action on the $\ell$-primary part of Abelian $3$-folds of Picard type

Abstract

Half a century ago Manin showed that given a number field $k$ and a rational prime $\ell$, there exists a uniform bound for the order of cyclic $\ell$-power isogenies between two non-CM elliptic curves over $k$. We generalize this to certain $2$-dimensional families of abelian $3$-folds with multiplication by an imaginary quadratic field.