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The Mordell–Weil sieve: proving non-existence of rational points on curves

Part of: Arithmetic problems. Diophantine geometry Curves Diophantine equations Arithmetic algebraic geometry Computational number theory Computational aspects in algebraic geometry

Published online by Cambridge University Press:  01 August 2010

Nils Bruin  and
Michael Stoll
Nils Bruin
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada (email: [email protected])
Michael Stoll
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany (email: [email protected])

Abstract

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We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.

Supplementary materials are available with this article.

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 11Y50: Computer solution of Diophantine equations 14G05: Rational points 14G25: Global ground fields 14H25: Arithmetic ground fields 14H45: Special curves and curves of low genus 14Q05: Curves
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 272 - 306
Copyright
Copyright © London Mathematical Society 2010

References

[1] Baker, A. and Wüstholz, G., Logarithmic forms and Diophantine geometry, New Mathematical Monographs 9 (Cambridge University Press, Cambridge, 2007).Google Scholar
[2] Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: The user language’, J. Symb. Comp. 24 (1997) 235–265. Also see the MAGMA home page at http://magma.maths.usyd.edu.au/magma/.CrossRefGoogle Scholar
[3] Bruin, N., ‘The arithmetic of Prym varieties in genus 3’, Compositio Math. 144 (2008) 317–338.CrossRefGoogle Scholar
[4] Bruin, N. and Elkies, N. D., ‘Trinomials ax 7+bx+c and ax 8+bx+c with Galois groups of order 168 and 8*168’, Algorithmic number theory: 5th international symposium, ANTS-V (Sydney, Australia, July 2002) proceedings, Lecture Notes in Computer Science 2369 (eds Fieker, Claus and Kohel, David R.; Springer, Berlin, 2002) 172–188.CrossRefGoogle Scholar
[5] Bruin, N. and Stoll, M., ‘Deciding existence of rational points on curves: an experiment’, Experiment. Math. 17 (2008) 181–189.CrossRefGoogle Scholar
[6] Bruin, N. and Stoll, M., ‘2-cover descent on hyperelliptic curves’, Math. Comp. 78 (2009) 2347–2370.CrossRefGoogle Scholar
[7] Bruin, N. and Stoll, M., ‘MWSieve-new.m’, MAGMA code for Mordell–Weil sieve computation, 2009, electronic appendix to ‘The Mordell–Weil sieve: proving non-existence of rational points on curves’, LMS J. Comput. Math. 13 (2010) 272–306, doi: 10.1112/S1461157009000187.CrossRefGoogle Scholar
[8] Bugeaud, Y., Mignotte, M., Siksek, S., Stoll, M. and Tengely, Sz., ‘Integral points on hyperelliptic curves’, Algebra Number Theory 2 (2008) 859–885.CrossRefGoogle Scholar
[9] Cantor, D. G., ‘Computing in the Jacobian of a hyperelliptic curve’, Math. Comp. 48 (1987) 95–101.CrossRefGoogle Scholar
[10] Cassels, J. W. S. and Flynn, E. V., Prolegomena to a middlebrow arithmetic of curves of genus 2 (Cambridge University Press, Cambridge, 1996).CrossRefGoogle Scholar
[11] Chabauty, C., ‘Sur les points rationnels des courbes algébriques de genre supérieur à l’unité’, C. R. Acad. Sci. Paris 212 (1941) 882–885 (in French).Google Scholar
[12] Coleman, R. F., ‘Effectve Chabauty’, Duke Math. J. 52 (1985) 765–770.CrossRefGoogle Scholar
[13] Flynn, E. V., ‘A flexible method for applying Chabauty’s theorem’, Compositio Math. 105 (1997) 79–94.CrossRefGoogle Scholar
[14] Flynn, E. V., ‘The Hasse principle and the Brauer–Manin obstruction for curves’, Manuscripta Math. 115 (2004) 437–466.CrossRefGoogle Scholar
[15] Flynn, E. V., FTP site with formulas relating to genus 2 curves, http://people.maths.ox.ac.uk/∼flynn/genus2/.Google Scholar
[16] Flynn, E. V. and Smart, N. P., ‘Canonical heights on the Jacobians of curves of genus 2 and the infinite descent’, Acta Arith. 79 (1997) 333–352.CrossRefGoogle Scholar
[17] Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics 52 (Springer, New York, 1977).CrossRefGoogle Scholar
[18] McCallum, W. and Poonen, B., The method of Chabauty and Coleman, Preprint, 2007, http://www-math.mit.edu/∼poonen/papers/chabauty.pdf. Proceedings of the 2004 IHP Trimestre on Explicit Methods in Number Theory, in the ‘Panorama & Syntheses’ series of the SMF, to appear.Google Scholar
[19] Murty, V. K. and Scherk, J., ‘Effective versions of the Chebotarev density theorem for function fields’, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994) 523–528.Google Scholar
[20] Pohlig, G. C. and Hellman, M. E., ‘An improved algorithm for computing logarithms over GF(p) and its cryptographic significance’, IEEE Trans. Inform. Theory IT-24 (1978) 106–110.CrossRefGoogle Scholar
[21] Poonen, B., ‘Heuristics for the Brauer–Manin obstruction for curves’, Experiment. Math. 15 (2006) 415–420.CrossRefGoogle Scholar
[22] Poonen, B., Schaefer, E. F. and Stoll, M., ‘Twists of X(7) and primitive solutions to x 2+y 3=z 7’, Duke Math. J. 137 (2007) 103–158.CrossRefGoogle Scholar
[23] Scharaschkin, V., ‘Local-global problems and the Brauer–Manin obstruction’, PhD Thesis, University of Michigan, 1999.Google Scholar
[24] Shafarevich (ed.), I. R., Algebraic geometry I, Encyclopaedia of Mathematical Sciences 23 (Springer, Berlin, 1994).CrossRefGoogle Scholar
[25] Stoll, M., ‘On the height constant for curves of genus two’, Acta Arith. 90 (1999) 183–201.CrossRefGoogle Scholar
[26] Stoll, M., ‘Implementing 2-descent on Jacobians of hyperelliptic curves’, Acta Arith. 98 (2001) 245–277.CrossRefGoogle Scholar
[27] Stoll, M., ‘On the height constant for curves of genus two, II’, Acta Arith. 104 (2002) 165–182.CrossRefGoogle Scholar
[28] Stoll, M., ‘Independence of rational points on twists of a given curve’, Compositio Math. 142 (2006) 1201–1214.CrossRefGoogle Scholar
[29] Stoll, M., ‘Finite descent obstructions and rational points on curves’, Algebra Number Theory 1 (2007) 349–391.CrossRefGoogle Scholar
[30] Stoll, M., ‘Applications of the Mordell–Weil sieve’, Oberwolfach Rep. 4 (2007) 1967–1970.Google Scholar
[31] Stoll, M., ‘How to obtain global information from computations over finite fields’, Higher-dimensional geometry over finite fields, NATO Science for Peace and Security Series: Information and Communication Security 16 (eds Kaledin, D. and Tschinkel, Y.; IOS Press, Amsterdam, 2008) 189–196.Google Scholar
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