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On elliptic curves of prime power conductor over imaginary quadratic fields with class number one
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Cremona, J. E. and Pacetti, Ariel (2019) On elliptic curves of prime power conductor over imaginary quadratic fields with class number one. Proceedings of the London Mathematical Society, 118 (5). pp. 1245-1276. doi:10.1112/plms.12214
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Official URL: https://doi.org/10.1112/plms.12214
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-Oesterlé (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.
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Curves, Elliptic, Quadratic fields | ||||||||||||
| Journal or Publication Title: | Proceedings of the London Mathematical Society | ||||||||||||
| Publisher: | Cambridge University Press | ||||||||||||
| ISSN: | 0024-6115 | ||||||||||||
| Official Date: | May 2019 | ||||||||||||
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| Volume: | 118 | ||||||||||||
| Number: | 5 | ||||||||||||
| Page Range: | pp. 1245-1276 | ||||||||||||
| DOI: | 10.1112/plms.12214 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
| RIOXX Funder/Project Grant: |
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