Timeline for An elliptic curve for Ramanujan-type cubic identities?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 5, 2019 at 10:08 | comment | added | davidoff303 | Here is similar question. | |
| Jul 6, 2017 at 14:42 | answer | added | davidoff303 | timeline score: 6 | |
| Jul 7, 2016 at 19:31 | vote | accept | Tito Piezas III | ||
| Jul 6, 2016 at 1:48 | comment | added | Michael Zieve | What is surely happening is the following: you must be getting two equations in the three variables $u,v,w$, which together define an elliptic curve. The image of this elliptic curve under the projection $(u,v,w)\mapsto (u,w)$ is your equation (2), and this projection is a $3$-isogeny. You get infinitely many rational $u,v,w$ because the point from your Solution 1 is a rational point on the $(u,v,w)$ curve having infinite order. | |
| Jul 5, 2016 at 15:45 | answer | added | Michael Zieve | timeline score: 8 | |
| Jul 5, 2016 at 7:36 | comment | added | Felipe Voloch | What you wrote is almost certainly a point of infinite order when you regard $p,q$ as variables. This will imply that, for most values of $p,q$, you will get a point of infinite order too but there will certainly be specializations (perhaps not rational) that will give a point of finite order. | |
| Jul 5, 2016 at 1:17 | comment | added | Tito Piezas III | @MichaelZieve: What I did was to use an initial solution $u_0$ to generate $u_1$, then $u_2$, and so on, each one becoming more complex. Then I constructed the nonic in $v$, plugged in the $u_i$, and observed if it had a rational root. So far, it has for several iterations already. So I am assuming I can do this ad infinitum, because it would be strange if it would suddenly stop at say, $u_8$. Also, I modified the sentence after $(2)$ re your comment. | |
| Jul 5, 2016 at 1:07 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
More details.
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| Jul 4, 2016 at 23:57 | comment | added | Michael Zieve | Do you know that there are infinitely many pairs of rational numbers $u,v$ satisfying your equation (1)? If so, what is the justification for that? Also, just a minor comment: your sentence after (2) sounds like you're saying that every elliptic curve over $\mathbf{Q}$ has infinitely many rational points, when instead you're presumably saying that the difference between the two "obvious" rational points (namely $(u,w)$ for the value of $u$ you described and the corresponding value of $w$, and the unique rational point at infinity) is a rational point in the Jacobian having infinite order. | |
| Jul 4, 2016 at 23:48 | history | edited | Myshkin |
+ top level tag (nt.)
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| Jul 4, 2016 at 21:42 | history | asked | Tito Piezas III | CC BY-SA 3.0 |