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

  • Nils Bruin (a1) and Michael Stoll (a2)
Abstract
Abstract

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.

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[4] N. Bruin and N. D. Elkies , ‘Trinomials ax7+bx+c and ax8+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 Claus Fieker and David R. Kohel ; Springer, Berlin, 2002) 172188.

[10] J. W. S. Cassels and E. V. Flynn , Prolegomena to a middlebrow arithmetic of curves of genus 2 (Cambridge University Press, Cambridge, 1996).

[17] R. Hartshorne , Algebraic geometry, Graduate Texts in Mathematics 52 (Springer, New York, 1977).

[24] I. R. Shafarevich (ed.) , Algebraic geometry I, Encyclopaedia of Mathematical Sciences 23 (Springer, Berlin, 1994).

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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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