joint with John Cremona, Catherine O'Neil, Denis Simon and Michael Stoll
This is the second in a series of papers in which we study the n-Selmer group of an elliptic curve. In this paper, we show how to realize elements of the n-Selmer group explicitly as curves of degree n embedded in Pn-1. The main tool we use is a comparison between an easily obtained embedding into Pn2-1 and another map into Pn2-1 that factors through the Segre embedding Pn-1 x Pn-1 -> Pn2-1. The comparison relies on an explicit version of the local-to-global principle for the n-torsion of the Brauer group of the base field.
The other papers in this series are Paper I. Algebra and Paper III. Algorithms.