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Explicit n-descent on elliptic curves, II. Geometry

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 P^n-1. The main tool we use is a comparison between an easily obtained embedding into P^n²-1 and another map into P^n²-1 that factors through the Segre embedding P^n-1 x P^n-1 -> P^n²-1. 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.

Explicit n-descent on elliptic curves, II. Geometry (24 pages)

The other papers in this series are Paper I. Algebra and Paper III. Algorithms.