Séminaire Combinatoire et Théorie des Nombres ICJ

Elliptic curves of prime conductor over imaginary quadratic fields of class number one

by John Cremona (University of Warwick)

Bât. Braconnier, salle Fokko du Cloux (ICJ, Université Lyon 1)

Bât. Braconnier, salle Fokko du Cloux

ICJ, Université Lyon 1


We extend from Q to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterlé (1989), namely that if E is an elliptic curve of prime conductor then either 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 (up to isogeny) 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 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.

This is joint work with Ariel Pacetti (Córdoba, Argentina)

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