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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Explicit n-descent on elliptic curves, II. Geometry

J. E. Cremona1 / T. A. Fisher2 / C. O'Neil3 / D. Simon4 / M. Stoll5

1Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK. e-mail:

2University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK. e-mail:

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

4Université de Caen, Campus II—Boulevard Maréchal Juin, BP 5186, 14032 Caen, France. e-mail:

5Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany. e-mail:

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

Publication History

Published Online:


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.

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