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Ranks of elliptic curves with prescribed torsion over number fields


Bosmans, Johan; Bruin, Peter; Dujella, Andrej; Najman, Filip (2014). Ranks of elliptic curves with prescribed torsion over number fields. International Mathematics Research Notices, 2014(11):2885-2923.

Abstract

We study the structure of Mordell-Weil groups of elliptic curves over number fields of degrees 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group T and a quartic field K such that among the elliptic curves over K with torsion subgroup T, there are curves of positive rank, but none of rank 0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call false complex multiplication, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization

Abstract

We study the structure of Mordell-Weil groups of elliptic curves over number fields of degrees 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group T and a quartic field K such that among the elliptic curves over K with torsion subgroup T, there are curves of positive rank, but none of rank 0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call false complex multiplication, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:National licences > 142-005
Dewey Decimal Classification:Unspecified
Language:English
Date:1 January 2014
Deposited On:02 Nov 2018 16:25
Last Modified:24 Sep 2019 23:41
Publisher:Oxford University Press
ISSN:1073-7928
OA Status:Green
Publisher DOI:https://doi.org/10.1093/imrn/rnt013
Related URLs:https://www.swissbib.ch/Search/Results?lookfor=nationallicenceoxford101093imrnrnt013 (Library Catalogue)

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