Vol. 4, No. 6, 2010

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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
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 , 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/