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Mathematics of Computation

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Sieving for rational points on hyperelliptic curves

Author: Samir Siksek
Journal: Math. Comp. 70 (2001), 1661-1674
MSC (2000): Primary 11G05; Secondary 11Y16, 11Y50
Published electronically: March 7, 2001
MathSciNet review: 1836925
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We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations $\mathbb{Q}_p$. We illustrate the practicality of the method with some examples of hyperelliptic curves of genus $1$.

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Additional Information

Samir Siksek
Affiliation: Institute of Mathematics and Statistics, Cornwallis Building, University of Kent, Canterbury, UK
Address at time of publication: Department of Mathematics, College of Science, PO Box 36, Sultan Qaboos University, Oman

Keywords: Diophantine equations, elliptic curves
Received by editor(s): November 21, 1996
Received by editor(s) in revised form: January 28, 1997, and November 29, 1999
Published electronically: March 7, 2001
Additional Notes: The author’s research was conducted while the author was at the University of Kent and funded by a grant from the EPSRC (UK).
Dedicated: To Shaheen
Article copyright: © Copyright 2001 American Mathematical Society