John Cremona

Elliptic Curves of prime power conductor over imaginary quadratic fields with class number one.

We generalize from Q to imaginary quadratic fields of class number one a result of Serre and Mestre-Oesterle of 1989, namely that if E is an elliptic curve of prime conductor then the isogeny class of E contains a curve with prime discriminant. The proof is conditional in two ways: the curves are assumed modular, so are associated to suitable Bianchi newforms; and also that a certain level-lowering conjecture holds for Bianchi newforms.

Along with the proof of this main result, we classify all elliptic curves of prime power conductor and non-trivial torsion over each of the fields in question: in the case of 2-torsion we find that apart from a very small number of sporadic cases, such curves either have CM or arise from a family analogous to the Setzer-Neumann family of elliptic curves over Q.

This is joint work with Ariel Pacetti (Cordoba, Argentina).