Minimisation and reduction of 5-coverings of elliptic curves

Fisher, Tom

doi:10.2140/ant.2013.7.1179

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Minimisation and reduction of 5-coverings of elliptic curves

Tom Fisher

Vol. 7 (2013), No. 5, 1179–1205

DOI: 10.2140/ant.2013.7.1179

Abstract

We consider models for genus- $1$ curves of degree $5$ , which arise in explicit $5$ -descent on elliptic curves. We prove a theorem on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and give an algorithm for computing such models. Finally we describe how to reduce genus- $1$ models of degree $5$ defined over $ℚ$ .

Keywords

elliptic curves, genus-$1$ curves, minimisation, reduction, descent

Mathematical Subject Classification 2010

Primary: 11G05

Secondary: 11G07, 14H52, 14H25

Milestones

Received: 2 February 2012

Accepted: 20 August 2012

Published: 6 September 2013

Authors

Tom Fisher
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom
http://www.dpmms.cam.ac.uk/~taf1000/

Vol. 7, No. 5, 2013