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Mark Watkins
A note on integral points on elliptic curves
Journal de théorie des nombres de Bordeaux, 18 no. 3 (2006), p. 707-720, doi: 10.5802/jtnb.568
Article PDF | Reviews MR 2330437 | Zbl 1124.11028 | 2 citations in Cedram

Résumé - 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.

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