Timeline for Division by 3 on elliptic curve
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 4, 2013 at 9:37 | comment | added | Enrique Gonzalez-Jimenez | @Michael: Thanks for the answer. @All: Yes, I didn't say nothing if I would like something explicit or not. Sorry. | |
| Mar 31, 2013 at 14:15 | comment | added | Michael Stoll | @Joe: The OP didn't say what he was needing the answer for. You're right, of course, for the application to computing MW ranks. But then, the same will be true for the arithmetic of $K$ in the case when $E[m] \subset E(K)$. | |
| Mar 30, 2013 at 17:40 | comment | added | Joe Silverman | @Michael: Sorry, you're right. Given a specific point, its divisibility by 3 (or by 2, or by $m$) is a purely algebraic question. However, usually the reason to do this is to compute the Mordell-Weil group, so (for $m=2$) one looks at equations $x-\alpha_i=y_i^3$ for $i=1,2,3$. And then the arithmetic of $L$ will come into play. For $X_1(11)$ and $m=2$, my recollection is that $L$ has class number 1 and that one can completely characterize the units that are squares, which was crucial in showing that $X_1(11)(\mathbb{Q})$ has rank $0$. | |
| Mar 29, 2013 at 22:18 | comment | added | Michael Stoll | @Joe: For the OP's question (how to determine if a point is divisible by 3), we only need to be able to check if an element of $L$ is a cube, which is a purely algebraic question and does not require any arithmetic information. (Take the minimal polynomial $p$ of your element and check if $p(x^3)$ has a root in $L$.) | |
| Mar 29, 2013 at 12:45 | comment | added | Joe Silverman | @Michael: I had assumed the OP was primarily interested in the situation where one can reduce to questions about cubes in $K$, but you're quite right that this is the right generalization if one doesn't assume that the 3-torsion is rational. Of course, now the arithmetic of $L$ will come into play, especially the class group and unit group. I first saw this used (for 2-torsion) in a letter that Tate wrote computed $E(Q)$ for $E=X_1(11)$. For general $m$, Bob Wake in his thesis partially generalized the idea of looking at $L^*/{L^*}^m$, but I don't think that he ever published his thesis. | |
| Mar 28, 2013 at 19:24 | history | answered | Michael Stoll | CC BY-SA 3.0 |