Vol. 7, No. 5, 2013

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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
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