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

TL;DR

This paper extends classical results about elliptic curves of prime conductor from rational numbers to nine imaginary quadratic fields with class number one, classifying their properties and torsion structures under certain conjectural assumptions.

Contribution

It generalizes Serre's and Mestre-Oesterl's results to nine imaginary quadratic fields, classifies elliptic curves of prime power conductor with torsion, and provides conditional proofs based on modularity and level-lowering conjectures.

Findings

For four fields, the theorem holds with no change.

For five fields, the discriminant is either prime or a prime square.

Classified all elliptic curves of prime power conductor with non-trivial torsion.

Abstract

The main result of this paper is to extend from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterl\'e (1989), namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-, $3$- or $5$-isogenous curve has prime discriminant. For four of the nine fields, the theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and second 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 over $\Q$.