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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

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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:
DateEvent
May 2019Published
29 October 2018Accepted
22 November 2018Available
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:
Project/Grant IDRIOXX Funder NameFunder ID
EP/K034383/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
#676541H2020 European Research Councilhttp://dx.doi.org/10.13039/100010663
PIP 2014-2016 11220130100073Leverhulme Trusthttp://dx.doi.org/10.13039/501100000275
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