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

John Cremona, Ariel Pacetti

Published 2017-11-06Version 1

The main result of this paper is to generalize from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre and Mestre-Oesterl\'e of 1989, namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-isogenous curve or a $3$-isogenous curve has prime discriminant. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and secondly that a certain level-lowering conjecture holds for Bianchi newforms. We also classify all elliptic curves of prime power conductor and non-trivial torsion over each of the nine fields: in the case of $2$-torsion we find that such curves either have CM or with a small (finite) number of exceptions arise from a family analogous to the Setzer-Neumann family of elliptic curves over $\Q$.