The Mordell-Weil sieve: proving non-existence of rational points on curves. (English) Zbl 1278.11069

Bruin, Nils; Stoll, Michael

LMS J. Comput. Math. 13, 272-306 (2010).

Summary: 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.

MSC:

11G30	Curves of arbitrary genus or genus \(\ne 1\) over global fields
11Y50	Computer solution of Diophantine equations
14G05	Rational points

Software:

Magma

References:

[1]	Stoll, Oberwolfach Rep. 4 pp 1967– (2007)
[2]	Stoll, Acta Arith. 90 pp 183– (1999)
[3]	Shafarevich (ed.), Algebraic geometry I (1994)
[4]	Cantor, Math. Comp. 48 pp 95– (1987)
[5]	Poonen, Experiment. Math. 15 pp 415– (2006) · Zbl 1173.11040
[6]	Murty, C. R. Acad. Sci. Paris Sér. I Math. 319 pp 523– (1994)
[7]	Stoll, Higher-dimensional geometry over finite fields pp 189– (2008)
[8]	Bruin, Algorithmic number theory: 5th international symposium, ANTS-V (Sydney, Australia, July 2002) proceedings pp 172– (2002)
[9]	Hartshorne, Algebraic geometry (1977) · Zbl 0367.14001
[10]	Flynn, Acta Arith. 79 pp 333– (1997)
[11]	Baker, Logarithmic forms and Diophantine geometry (2007) · Zbl 1145.11004
[12]	Chabauty, C. R. Acad. Sci. Paris 212 pp 882– (1941)
[13]	Cassels, Prolegomena to a middlebrow arithmetic of curves of genus 2 (1996) · Zbl 0857.14018
[14]	Bruin, Experiment. Math. 17 pp 181– (2008) · Zbl 1218.11065