A note on integral points on elliptic curves

Watkins, Mark

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 707-720.

À la suite de Zagier et Elkies, nous recherchons de grands points entiers sur des courbes elliptiques. En écrivant une solution polynomiale générique et en égalisant des coefficients, nous obtenons quatre cas extrémaux susceptibles d’avoir des solutions non dégénérées. Chacun de ces cas conduit à un système d’équations polynomiales, le premier ayant été résolu par Elkies en 1988 en utilisant les résultants de Macsyma ; il admet une unique solution rationnelle non dégénérée. Pour le deuxième cas nous avons constaté que les résultants ou les bases de Gröbner sont peu efficaces. Suivant une suggestion d’Elkies, nous avons alors utilisé une itération de Newton $p$-adique multidimensionnelle et découvert une solution non dégénérée, quoique sur un corps de nombres quartique. En raison de notre méthodologie, nous avons peu d’espoir de montrer qu’il n’y a aucune autre solution. Pour le troisième cas nous avons trouvé une solution sur un corps de degré 9, mais n’avons pu traiter le quatrième cas. Nous concluons par quelques commentaires et une annexe d’Elkies concernant ses calculs et sa correspondance avec Zagier.

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.

[1] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. In Computational algebra and number theory Proceedings of the 1st MAGMA Conference held at Queen Mary and Westfield College, London, August 23–27, 1993. Edited by J. Cannon and D. Holt, Elsevier Science B.V., Amsterdam (1997), 235–265. Cross-referenced as J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Online at http://magma.maths.usyd.edu.au | MR | Zbl

[2] R. P. Brent, Algorithms for Minimization Without Derivatives. Prentice-Hall, Englewood Cliffs, NJ, 1973. | MR | Zbl

[3] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations. Math. Comp. 19 (1965), 577–593. | MR | Zbl

[4] H. Cohen, A course in computational algebraic number theory. Graduate Texts in Mathematics, 138. Springer-Verlag, New York, 1993. | MR | Zbl

[5] N. D. Elkies, Shimura curves for level-3 subgroups of the (2,3,7) triangle group, and some other examples. To appear in ANTS-VII proceedings, online at | arXiv | MR

[6] N. D. Elkies, M. Watkins, Polynomial and Fermat-Pell families that attain the Davenport-Mason bound. In progress.

[7] M. J. Greenberg Lectures on forms in many variables. W. A. Benjamin, Inc., New York-Amsterdam, 1969. | MR | Zbl

[8] M. Hall Jr., The Diophantine equation $x^3-y^2=k$. In Computers in number theory, Proceedings of the Science Research Council Atlas Symposium No. 2 held at Oxford, from 18–23 August 1969. Edited by A. O. L. Atkin and B. J. Birch. Academic Press, London-New York (1971), 173–198. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[13] D. Zagier, Large Integral Points on Elliptic Curves, and addendum. Math. Comp. 48 (1987), no. 177, 425–436, 51 (1988), no. 183, 375. | MR | Zbl