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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 $1$.

References [Enhancements On Off] (What's this?)

  • [AHU] A. V. Aho, J. E. Hopcroft, J. D. Ullman, Data Structures and Algorithms, Addison-Wesley, 1982. MR 84f:68001
  • [Cal] J. W. S. Cassels, Local Fields, LMS Student Texts, Cambridge University Press, 1986. MR 87i:11172
  • [Ca2] J. W. S. Cassels, Survey Article: Diophantine Equations with Special Reference to Elliptic Curves, J.L.M.S. 41 (1966), 193-291. MR 33:7299
  • [Ca3] J. W. S. Cassels, Second Descents for Elliptic Curves, J. reine angew. Math. 494 (1998), 101-127. MR 99d:11058
  • [Cohen] H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer-Verlag, third corrected printing, 1996. MR 94i:11105
  • [Cohn] P. M. Cohn, Algebra, Volume I, second edition, John Wiley and Sons, 1982. MR 83e:00002
  • [Cre1] J. E. Cremona, Algorithms for Modular Elliptic Curves, second edition, Cambridge University Press, 1997. MR 99e:11068
  • [Cre2] J. E. Cremona, Personal Communication, 1996.
  • [Me,Si,Sm] J.R. Merriman, S. Siksek and N.P. Smart, Explicit 4-Descents on an Elliptic Curve, Acta Arith. LXXVII (1996), 385-404. MR 97j:11027
  • [Sil] J. H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag, 1986. MR 87g:11070

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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

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