The Library
On elliptic curves of prime power conductor over imaginary quadratic fields with class number one
Tools
Tools
Tools
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 ISSN 0024-6115.
|
PDF
WRAP-elliptic-curves-prime-power-quadratic-class-one-Cremona-2018.pdf - Accepted Version - Requires a PDF viewer. Download (811Kb) | Preview |
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 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | ||||||||||||
| Dates: |
|
||||||||||||
| 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 | ||||||||||||
| Date of first compliant deposit: | 5 November 2018 | ||||||||||||
| Date of first compliant Open Access: | 6 November 2018 | ||||||||||||
| RIOXX Funder/Project Grant: |
|
||||||||||||
| Related URLs: | |||||||||||||
| Open Access Version: |
Request changes or add full text files to a record
Repository staff actions (login required)
 |
View Item |
Downloads
Downloads per month over past year
