Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves

Cremona, John E.; Fisher, Tom A.; Stoll, Michael

doi:10.2140/ant.2010.4.763

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Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves

John E. Cremona, Tom A. Fisher and Michael Stoll

Vol. 4 (2010), No. 6, 763–820

DOI: 10.2140/ant.2010.4.763

Abstract

We consider models for genus-one curves of degree $n$ for $n = 2$ , $3$ and $4$ , which arise in explicit $n$ -descent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general number fields. Finally, for genus-one models defined over $ℚ$ , we develop a theory of reduction and again give explicit algorithms for $n = 2$ , $3$ and $4$ .

Keywords

elliptic curves, genus-one curves, minimisation, reduction, descent

Mathematical Subject Classification 2000

Primary: 11G05

Secondary: 11G07, 11G05, 14H52, 14H25

Milestones

Received: 19 January 2010

Accepted: 18 July 2010

Published: 25 September 2010

Authors

John E. Cremona
Mathematics Institute Zeeman Building University of Warwick Coventry CV4 7AL United Kingdom
http://www.warwick.ac.uk/staff/J.E.Cremona
Tom A. Fisher
DPMMS, Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom
http://www.dpmms.cam.ac.uk/~taf1000/
Michael Stoll
Universität Bayreuth Mathematisches Institut 95440 Bayreuth Germany
http://www.mathe2.uni-bayreuth.de/stoll/

Vol. 4, No. 6, 2010