Timeline for Algorithms for finding rational points on an elliptic curve?
Current License: CC BY-SA 2.5
7 events
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| Jan 9, 2018 at 16:44 | comment | added | William Stein | I would guess not. People just tried computing the MW group of $y^2 = x^3 + k$ for all positive $k<10000$, and succeeded, except for $k=7823$, which required new methods to complete. | |
| Jan 8, 2018 at 15:21 | comment | added | Tito Piezas III | @WilliamStein: I know this is an old comment, but I find your rank 1 curve $$y^2 = x^3+7823 = x^3+(48\times163-1)$$ very intriguing. Does the $163$ have anything to do with it? | |
| Nov 20, 2010 at 21:28 | comment | added | William Stein | Here's an example of finding a point on a rank one curve in Sage using Heegner points: "E = EllipticCurve('37a'); P = E.heegner_point(-7); P.point_exact()". The rank 1 curve $y^2=x^3+7823$ provides a spectacular example in which Heegner points fail in practice, but doing a FOUR descent succeeds. Michael Stoll wrote a paper about this, but I can't find it online anymore (it vanished from his website), so here is a temporary link: wstein.org/home/wstein/tmp/4-descent.pdf Also, having read the ancient Peter Green Heegner points program you link above, I do not recommend it. | |
| Nov 20, 2010 at 21:20 | comment | added | William Stein | A very recent version of ratpoints is in Sage. Inputing "import sage.libs.ratpoints as r; r.ratpoints([46224, -3024, 0, 1], 200)" will output lots of points on the curve $y^2 = x^3 - 3024x + 46224$, such as [(1, 0, 0), (-60, 108, 1), (-60, -108, 1), (-32, 332, 1), ...] | |
| Oct 13, 2010 at 17:02 | comment | added | Chris Wuthrich | Finally one should add that it is easy to find the torsion points, if there are any. Most of this is implemented in magma and lots of it in sage. | |
| Oct 13, 2010 at 17:01 | comment | added | Chris Wuthrich | I agree best with this answer. The descents (as in Robin's answer) tell us that in order to find rational points on an elliptic curve, we better search on one of its torsors. But in the end, we have to do some "brutal search" and that is where the crucial improvements in ratpoints are useful. ... and the only other known method to find rational points is by modularity, say by using Heegner points or variants of them, or (as Pollack and Kurihara do) using supersingular Iwasawa theory. But all of them only work when the analytic rank is 1. | |
| Oct 13, 2010 at 16:06 | history | answered | Felipe Voloch | CC BY-SA 2.5 |