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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Second $ p$-descents on elliptic curves


Author: Brendan Creutz
Journal: Math. Comp. 83 (2014), 365-409
MSC (2010): Primary 11G05, 11Y50
Published electronically: May 23, 2013
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Abstract: Let $ p$ be a prime and $ C$ a genus one curve over a number field $ k$ representing an element of order dividing $ p$ in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of $ D$ in the Shafarevich-Tate group such that $ pD = C$ and obtains explicit models for these $ D$ as curves in projective space. This leads to a practical algorithm for performing explicit $ 9$-descents on elliptic curves over $ \mathbb{Q}$.


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Additional Information

Brendan Creutz
Affiliation: School of Mathematics and Statistics, Carslaw Building F07, University of Sydney, NSW 2006, Australia
Email: brendan.creutz@sydney.edu.au

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02713-5
PII: S 0025-5718(2013)02713-5
Received by editor(s): August 27, 2011
Received by editor(s) in revised form: April 3, 2012, and April 23, 2012
Published electronically: May 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.