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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?

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4 Answers 4

There are precisely two available "serious" implementations of the standard algorithm for computing integral points on an elliptic curve: a non-free one in Magma (http://magma.maths.usyd.edu.au/magma/) and a free one in Sage (http://sagemath.org). The one in Sage was done by Cremona and two German masters students a few years ago, and when refereeing the Sage code, I compared the answers with Magma, and uncovered and reported numerous bugs in Magma, which were subsequently fixed. Here's how to use Sage to find all integral (or S-integral!) points on a curve over Q:

sage: E = EllipticCurve([1,2,3,4,5])
sage: E.integral_points()
[(1 : 2 : 1)]
sage: E.S_integral_points([2])
[(-103/64 : -233/512 : 1), (1 : 2 : 1)]

and here is how to use Magma:

> E := EllipticCurve([1,2,3,4,5]);
> IntegralPoints(E);
[ (1 : 2 : 1) ]
> SIntegralPoints(E, [2]);                                                     
[ (1 : 2 : 1), (-103/64 : -233/512 : 1) ]

Note that in both cases by default the points are only returned up to sign. In Sage you get both signs like this:

sage: E.integral_points(both_signs=True)
[(1 : -6 : 1), (1 : 2 : 1)]

Finally, you can use Magma for free online here: http://magma.maths.usyd.edu.au/calc/ and you can use Sage free here: http://demo.sagenb.org/. With Sage, you can also just download it for free and install it on your computer. With Magma, you have to pay between $100 and a few thousand dollars, depending on who you are, and deal with copy protection.

NOTE: Technically a system called SIMATH (http://tnt.math.se.tmu.ac.jp/simath/) had an implementation of computing integral points. But it was killed by our friends at Siemens Corp.

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I forgot to mention that the integral_points command in Sage is documented here: sagemath.org/doc/reference/sage/schemes/elliptic_curves/… –  William Stein Nov 20 '10 at 21:55
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WARNING: Once the Magma integral points code was fixed, the Magma developers found bugs in the Sage integral_points code, as of Sage-4.6.1. See trac.sagemath.org/sage_trac/ticket/10152 –  William Stein Mar 7 '11 at 0:01
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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.

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The non-Weierstrass form is taken care of in Stroeker and de Weger's "Solving elliptic diophantine equations: The general cubic case." I recently had cause to work through it carefully, and the trick to deal with the non integrality-preservingness of the transform is pretty slick. –  Cam McLeman Sep 27 '10 at 1:01
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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.

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Chapter 5 of Silverman-Tate is about integral points, and about why there are finitely many. It doesn't give much help in actually determining them, which was the question. –  William Stein Nov 20 '10 at 21:41
    
Oh okay. It's been a while since I've worked with that book. Thanks for the clarification. –  Aaron Mazel-Gee Nov 21 '10 at 19:17
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Lemermeyer's resources list of links etc. on elliptic curves.

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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 '09 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 '09 at 21:18
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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 '10 at 20:46
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