EXPLICIT n-DESCENT ON ELLIPTIC CURVES II. GEOMETRY

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×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.