A note on integral points on elliptic curves
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.

@article{JTNB_2006__18_3_707_0,
     author = {Watkins, Mark},
     title = {A note on integral points on elliptic curves},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {707--720},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {3},
     year = {2006},
     doi = {10.5802/jtnb.568},
     mrnumber = {2330437},
     zbl = {1124.11028},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.568/}
}
Watkins, Mark. A note on integral points on elliptic curves. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 707-720. doi : 10.5802/jtnb.568. http://www.numdam.org/articles/10.5802/jtnb.568/

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