Timeline for Is elliptic curve point division defined over the field of real numbers?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 26, 2014 at 22:05 | vote | accept | Rich Apodaca | ||
| Nov 20, 2014 at 19:14 | comment | added | Joe Silverman | @RichApodaca The process is very feasible, in the following sense. I'll describe one case. Often $E(\mathbb{R})$ is isomorphic as a group to $\mathbb{R}^*/q^{\mathbb Z}$ for a real $q$ that's easy to compute to lots of decimal places. Further, the isomorphism is given by an explicit, rapidly converging series. So we have an isomorphism $f:\mathbb{R}^*/q^{\mathbb Z}\to E(\mathbb{R})$. You're taking a point $P\in E(\mathbb{R})$ and asking to solve $f(nt)=P$. So it's really a problem of numerical analysis: how easily can one solve such equations for an explicit power series $f(t)$? | |
| Nov 20, 2014 at 18:30 | comment | added | Rich Apodaca | @ChrisWuthrich, sage's division_points function was helpful (although I'm not a sage user). I'm not so much interested in finding a procedure as understanding how feasible the process is. For example, how does division_points perform as n and the coordinates of P increase in value? | |
| Nov 20, 2014 at 18:18 | comment | added | Rich Apodaca | @FelipeVoloch, thanks for the insight. What are the options for dividing by an odd number? | |
| Nov 20, 2014 at 17:21 | comment | added | Chris Wuthrich | If you are looking for "procedures" to compute it for general fields, then look at sage's function P.division_points(n). It starts by finding roots of the division polynomial in the field. | |
| Nov 20, 2014 at 17:20 | review | Close votes | |||
| Nov 22, 2014 at 16:39 | |||||
| Nov 20, 2014 at 17:07 | answer | added | Joe Silverman | timeline score: 10 | |
| Nov 20, 2014 at 17:02 | comment | added | Felipe Voloch | Over the reals there are elliptic logarithms and elliptic exponentials that convert the problem to division in real numbers. The only issue is that you can only divide by 2 (or any even number) if you are in the connected component of the identity. This is not an MO question, though. | |
| Nov 20, 2014 at 16:37 | review | First posts | |||
| Nov 20, 2014 at 17:24 | |||||
| Nov 20, 2014 at 16:35 | history | asked | Rich Apodaca | CC BY-SA 3.0 |