J. E. Cremona^{1} / T. A. Fisher^{2} / C. O'Neil^{3} / D. Simon^{4} / M. Stoll^{5}

^{1}Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK. e-mail: (email)

^{2}University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK. e-mail: (email)

^{3}Barnard College, Columbia University, Department of Mathematics, 2990 Broadway MC 4418, New York, NY 10027-6902, USA. e-mail: (email)

^{4}Université de Caen, Campus II—Boulevard Maréchal Juin, BP 5186, 14032 Caen, France. e-mail: (email)

^{5}Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany. e-mail: (email)

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2009, Issue 632, Pages 63–84, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2009.050, June 2009

## 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^{n2–1} and another map into ℙ^{n2–1} that factors through the Segre embedding ℙ^{n–1} × 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.

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