A note on integral points on elliptic curves
| dc.creator | Watkins, Mark | |
| dc.creator | Elkies, Noam D. | |
| dc.date | 2006-04-05 | |
| dc.date.accessioned | 2026-07-07T12:50:38Z | |
| dc.date.available | 2026-07-07T12:50:38Z | |
| dc.description | 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. | |
| dc.description | Submitted to the Pohst Birthday volume of the Journal de Théorie des Nombres de Bordeaux. With an appendix by Noam D. Elkies | |
| dc.identifier | https://arxiv.org/abs/math/0604097 | |
| dc.identifier | http://arxiv.org/abs/math/0604097 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/222741 | |
| dc.subject | Number Theory | |
| dc.title | A note on integral points on elliptic curves | |
| dc.type | text |