#### 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
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