|
Sieving for rational points on hyperelliptic curves
Author(s):
Samir
Siksek.
Journal:
Math. Comp.
70
(2001),
1661-1674.
MSC (2000):
Primary 11G05;
Secondary 11Y16, 11Y50
Posted:
March 7, 2001
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
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  . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus  .
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
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(2000):
11G05,
11Y16, 11Y50
Retrieve articles in all Journals with MSC
(2000):
11G05,
11Y16, 11Y50
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
Email:
siksek@squ.edu.om
DOI:
10.1090/S0025-5718-01-01275-3
PII:
S 0025-5718(01)01275-3
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
Posted:
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
Copyright of article:
Copyright
2001,
American Mathematical Society
|
Comments: webmaster@ams.org