Let A be an abelian threefold defined over a number field K whose endomorphism algebra is isomorphic to an imaginary quadratic field M. In recent joint work with Fité, we proved the existence of an elliptic curve E defined over K with CM by M such that for any prime \ell, the twisted Tate module V_\ell(E) (1)  is a sub representation of \wedge^3 V_\ell(A).

In this talk I will give an overview of the proof of the above result and present work in progress with Chidambaram and Fité where we provide explicit families of examples of the above phenomenon.