Abstract: Let A be an abelian threefold defined over a number field K with geometric endomorphism algebra an imaginary quadratic field M. In this talk, we will discuss the endomorphism structure of A (minimal field of definition for the endomorphisms and their action on the regular differentials) and show how to attach an elliptic curve E/K with CM by M to A whose associated Galois representations are determined by those of A.
As a corollary, we deduce that the class number of M is bounded by [K : Q]. This is joint work with Francesc Fité.