Timeline for Division by 3 on elliptic curve
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 28, 2013 at 21:06 | comment | added | Joe Silverman | @Enrique: I didn't say it was explicit, I said it's where the result comes from. In principle, one should be able to make it explicit by writing down explicit functions F, G and H, but first you'd need to write the elliptic curve in a form where its $m$-torsion is rational. So there's a fair amount of work involved in doing this, and I don't recall seeing it. But at least for $m=3$, it's likely to be feasible. | |
| Mar 28, 2013 at 19:17 | vote | accept | Enrique Gonzalez-Jimenez | ||
| Mar 28, 2013 at 19:17 | |||||
| Mar 28, 2013 at 18:09 | comment | added | Enrique Gonzalez-Jimenez | @Joe Silverman. Thanks for the answer. Although it seems that it is not very explicit. | |
| Mar 28, 2013 at 18:08 | vote | accept | Enrique Gonzalez-Jimenez | ||
| Mar 28, 2013 at 18:09 | |||||
| Mar 26, 2013 at 23:55 | comment | added | Joe Silverman | @wccanard: That was my first thought, too. Except it's not quite right, because if one of the three quantities is 0, then you really do need to check that the other two are squares. So my answer isn't quite right, either. It will work as long as P is not itself an m-torsion point, but if F(P) or G(P) vanishes, then there will be a third condition H(P) that needs to be checked. I believe that the ambiguity comes from making a specific choice of an isomorphism $E[m]\cong\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}$. | |
| Mar 26, 2013 at 22:46 | comment | added | user30035 | [Clarifying remark, perhaps obvious anyway: in the original question, the $m=2$ case, it looks like you have three functions which you need to check are squares, but if any two of them are squares then the third is automatically because their product is a square. So Joe's argument really does generalise the $m=2$ case above.] | |
| Mar 26, 2013 at 21:19 | history | answered | Joe Silverman | CC BY-SA 3.0 |