A note on integral points on elliptic curves

Mark Watkins[1]

  • [1] Department of Mathematics University Walk University of Bristol Bristol, BS8 1TW England

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 3, page 707-720
  • ISSN: 1246-7405

Abstract

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We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gröbner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional p -adic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodology, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth case. We make a few concluding comments and include an appendix from Elkies regarding his calculations and correspondence with Zagier.

How to cite

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Watkins, Mark. "A note on integral points on elliptic curves." Journal de Théorie des Nombres de Bordeaux 18.3 (2006): 707-720. <http://eudml.org/doc/249658>.

@article{Watkins2006,
abstract = {We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gröbner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional $p$-adic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodology, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth case. We make a few concluding comments and include an appendix from Elkies regarding his calculations and correspondence with Zagier.},
affiliation = {Department of Mathematics University Walk University of Bristol Bristol, BS8 1TW England},
author = {Watkins, Mark},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {elliptic curves; integral points},
language = {eng},
number = {3},
pages = {707-720},
publisher = {Université Bordeaux 1},
title = {A note on integral points on elliptic curves},
url = {http://eudml.org/doc/249658},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Watkins, Mark
TI - A note on integral points on elliptic curves
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 3
SP - 707
EP - 720
AB - We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gröbner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional $p$-adic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodology, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth case. We make a few concluding comments and include an appendix from Elkies regarding his calculations and correspondence with Zagier.
LA - eng
KW - elliptic curves; integral points
UR - http://eudml.org/doc/249658
ER -

References

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  9. S. Lang, Conjectured Diophantine estimates on elliptic curves. In Arithmetic and geometry. Vol. I., edited by M. Artin and J. Tate, Progr. Math., 35, Birkhäuser Boston, Boston, MA (1983), 155–171. Zbl0529.14017MR717593
  10. Macsyma, a sophisticated computer algebra system. See http://maxima.sourceforge.net for history and current version of its descendants. 
  11. PARI/GP, CVS development version 2.2.11, Université Bordeaux I, Bordeaux, France, June 2005. Online at http://pari.math.u-bordeaux.fr 
  12. P. Vojta, Diophantine approximations and value distribution theory. Lecture Notes in Mathematics, 1239. Springer-Verlag, Berlin, 1987. x+132 pp. Zbl0609.14011MR883451
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