On the ranks of the 2-Selmer groups of twists of a given elliptic curve

Kane, Daniel

doi:10.2140/ant.2013.7.1253

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On the ranks of the $2$-Selmer groups of twists of a given elliptic curve

Daniel M. Kane

Vol. 7 (2013), No. 5, 1253–1279

DOI: 10.2140/ant.2013.7.1253

Abstract

Swinnerton-Dyer considered the proportion of twists of an elliptic curve with full $2$ -torsion that have $2$ -Selmer group of a particular dimension. Swinnerton-Dyer obtained asymptotic results on the number of such twists using an unusual notion of asymptotic density. We build on this work to obtain similar results on the density of twists with particular rank of $2$ -Selmer group using the natural notion of density.

Keywords

Selmer group, elliptic curve, density

Mathematical Subject Classification 2010

Primary: 11G05

Milestones

Received: 27 June 2012

Revised: 4 January 2013

Accepted: 6 January 2013

Published: 6 September 2013

Authors

Daniel M. Kane
Stanford University Department of Mathematics Building 380, Sloan Hall Stanford, CA 94305 United States

Vol. 7, No. 5, 2013