Explicit n-descent on elliptic curves, II. Geometry

Abstract

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 P^{n-1}. The main tool we use is a comparison between an easily obtained embedding into P^{n^2-1} and another map into P^{n^2-1} that factors through the Segre embedding P^{n-1} x P^{n-1} --> P^{n^2-1}. 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.