Timeline for Local parameter at torsion points of elliptic curve
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 4, 2017 at 18:37 | comment | added | Michael Stoll | @User0829 $-x/y$ is certainly not canonical, as $x$ and $y$ themselves are not. But if you want a uniformizer of a simple form, then (a constant multiple of) $x/y$ is the simplest function that vanishes to order 1 at infinity that you can come up with. The minus sign makes the standard invariant differential look like $(1 + \dots) dt$, when $t = -x/y$. | |
| Mar 4, 2017 at 18:31 | comment | added | Michael Stoll | @JoeSilverman Sorry, I had somehow overlooked the "odd prime" requirement. | |
| Mar 4, 2017 at 14:28 | comment | added | User0829 | Moreover, is there a canonical way of choosing local parameters $\pi_O$ and $\pi_P$ at $O$ and $P_{\neq O}\in E[\ell]$ respectively so that $\pi_O\mapsto \pi_P$ under the induced ring homomorphism $\mathcal{O}_O\to\mathcal{O}_P$ from the $\ell$-isogeny $E\xrightarrow{[\ell]}E$. | |
| Mar 4, 2017 at 14:24 | comment | added | User0829 | Thank you very much. In fact, I have one more (stupid) question: I have read many literature choosing $-x/y$ as a local parameter at $O$. Is there any particular reason for that? I mean, is $-x/y$ `canonical' in some sense? | |
| Mar 4, 2017 at 13:34 | vote | accept | User0829 | ||
| Mar 3, 2017 at 22:29 | comment | added | Joe Silverman | @MichaelStoll Very true, and more generally if $P$ is a point of order $m$ with $m$ even, one has to be more careful. But the question specified that $\ell$ is a prime that is at least $3$. | |
| Mar 3, 2017 at 22:01 | comment | added | Michael Stoll | If $P$ is of order 2, then one should take $y$ instead (since then $x-a$ has order 2 at $P$). | |
| Mar 3, 2017 at 20:30 | history | answered | Joe Silverman | CC BY-SA 3.0 |