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Explicit $ n$-descent on elliptic curves III. Algorithms


Authors: J. E. Cremona, T. A. Fisher, C. O’Neil, D. Simon and M. Stoll
Journal: Math. Comp. 84 (2015), 895-922
MSC (2010): Primary 11G05, 14H25, 14H52
Published electronically: July 29, 2014
MathSciNet review: 3290968
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Abstract: This is the third in a series of papers in which we study the $ n$-Selmer group of an elliptic curve, with the aim of representing its elements as curves of degree $ n$ in  $ \mathbb{P}^{n-1}$. The methods we describe are practical in the case $ n=3$ for elliptic curves over the rationals, and have been implemented in MAGMA.

One important ingredient of our work is an algorithm for trivialising central simple algebras. This is of independent interest; for example, it could be used for parametrising Brauer-Severi surfaces.


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Additional Information

J. E. Cremona
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: J.E.Cremona@warwick.ac.uk

T. A. Fisher
Affiliation: University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: T.A.Fisher@dpmms.cam.ac.uk

C. O’Neil
Affiliation: 505 Pulitzer Hall, Columbia University Graduate School of Journalism, 2950 Broadway, New York, New York 10027
Email: cathy.oneil@gmail.com

D. Simon
Affiliation: Université de Caen, Campus II - Boulevard Maréchal Juin, BP 5186–14032, Caen, France
Email: Denis.Simon@math.unicaen.fr

M. Stoll
Affiliation: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
Email: Michael.Stoll@uni-bayreuth.de

DOI: https://doi.org/10.1090/S0025-5718-2014-02858-5
Received by editor(s): August 31, 2012
Received by editor(s) in revised form: June 4, 2013, and June 28, 2013
Published electronically: July 29, 2014
Article copyright: © Copyright 2014 American Mathematical Society