Siksek, Samir. (2001) Sieving for rational points on hyperelliptic curves. Mathematics of Computation, Vol.70 (No.236). pp. 1661-1674. ISSN 0025-5718

	PDF WRAP_Siksek_sieving.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (231Kb)
	PDF WRAP_siksek_coversheet.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (37Kb)

Abstract

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 Qp. We illustrate the practicality of the method with some examples of hyperelliptic curves of genus 1.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Sieves (Mathematics), Elliptic functions, Rational points (Geometry)
Journal or Publication Title:	Mathematics of Computation
Publisher:	American Mathematical Society
ISSN:	0025-5718
Date:	2001
Volume:	Vol.70
Number:	No.236
Page Range:	pp. 1661-1674
Identification Number:	10.1090/S0025-5718-01-01275-3
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	[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
URI:	http://wrap.warwick.ac.uk/id/eprint/4299

Request changes to a record

Actions (login required)

View Item