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Where can I found some resources to learn how to determine the integer points of given elliptic curve? I would like to learn a method based on computing the rank and the torsion group of given curve. Also, how can I determine the integer points if the curve is not on its Weierstrass form?

flag	asked Nov 24 at 11:28 Student 43●3

Aaron Mazel-Gee · Answer 1 · 2009-11-24 11:40:21Z UTC

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I would recommend Silverman & Tate's "Rational Points on Elliptic Curves", I'm pretty sure you'll find what you're looking for there.

answered Nov 24 at 11:40

Aaron Mazel-Gee
972●9

Franz Lemmermeyer · Answer 2 · 2010-02-21 20:38:14Z UTC

Just in case anyone is still reading: the tool of the trade are elliptic logarithms. A lot of people have worked on making this effective, but two recent articles pointing you in the right direction are

A- Pethö, H.-G. Zimmer, J. Gebel, E. Herrmann, Computing all $S$-integral points on elliptic curves, Math. Proc. Camb. Philos. Soc. 127 (1999), No.3, 383-402
R.J. Stroeker, N. Tzanakis, Computing all integer solutions of a genus 1 equation Math. Comput. 72 (2003), No. 244, 1917-1933

Gebel, Pethö and Zimmert have used this method for finding all integer points on Bachet-Mordell curves $y^2 = x^3+k$ for all small values of $k$, for example.

There are problems ahead if the curve is not in Weierstrass form since the transformation from a genus 1 curve to a curve in Weierstrass form does not preserve integrality. I do not remember whether you can find anything useful in the textbook

S. Schmitt, H.-G. Zimmer, Elliptic curves. A computational approach , de Gruyter (2003)

but would be surprised if you couldn't.

Thomas Riepe · Answer 3 · 2009-11-24 12:07:28Z UTC

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Lemermeyer's resources list of links etc. on elliptic curves.

answered Nov 24 at 12:07

Thomas Riepe
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	by the by, is Lemermeyer still active in his mathematics? I haven't heard much about him since 2004 or so. – Jose Capco Nov 24 at 14:06
	His website links to a school where he apparently teaches, more i don't know. BTW, could one read your "How to play like Alekhine"? Some of my friends are excellent chess players and surely would like to read it. – Thomas Riepe Nov 24 at 21:18
	Yes, I'm currently teaching at a gymnasium (the German equivalent of a high school). I spent a lot of my time since 2003 on projects in German (correspondence Hasse - Noether with Roquette, Jacobi's lectures in number theory from 1836/37 with H. Pieper). – Franz Lemmermeyer Feb 21 at 20:46

Integer points of an elliptic curve

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