Timeline for Ramification in p-division fields associated to elliptic curves with good ordinary reduction
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 15, 2012 at 13:31 | comment | added | Felipe Voloch | @Álvaro: I would expect it to apply to all curves, but it's been a while since I looked at this stuff. | |
| Mar 15, 2012 at 3:25 | comment | added | Álvaro Lozano-Robledo | @Felipe: Gross seems to assume (on p. 514) that the representation on the $p$-torsion is irreducible, so the criterion would not apply the curve 1225h1 listed in my original post, because it has a $\mathbb{Q}$-rational $37$-isogeny. Does the criterion apply to all elliptic curves with good reduction, or only to those with irreducible mod p representation? | |
| Jul 13, 2011 at 13:47 | vote | accept | Álvaro Lozano-Robledo | ||
| Jul 13, 2011 at 13:47 | comment | added | Álvaro Lozano-Robledo | Thanks! That was my best guess, that I was not understanding this correctly. I think I got it working now, and will post something else later. Thanks again. | |
| Jul 12, 2011 at 11:22 | comment | added | Felipe Voloch | @Álvaro: Careful, $s(x)$ is not $x^2$, only congruent to it modulo $2$. It is the Frobenius on Witt vectors which squares the Witt coordinates. I doubt there is a typo, Serre would have picked it up. Most of the corrections I have listed on that page were his. | |
| Jul 12, 2011 at 4:09 | comment | added | Álvaro Lozano-Robledo | Ok, if I am understanding everything correctly, there is a typo in the very last line of Lubin-Serre-Tate. When $p=2$ and $\lambda=1$, one should have $j_0 = 3375/31 = 3^3\cdot 5^3/31$. The polynomial $\phi_2(x,x^2)$ factors as $2^4(31x-3375)(3x^4 + 3x^3 + 82531x^2 + 26622000x + 2916000000)$, where $\phi_2(x,y)$ is the classical modular polynomial for $N=2$. The only root in $\mathbb{Q}_2$ congruent to $1$ mod $2$ is $3375/31$. | |
| Jul 9, 2011 at 14:22 | comment | added | Álvaro Lozano-Robledo | Thank you, Felipe! I will try to calculate $j_0$ in this case and report back. | |
| Jul 8, 2011 at 23:41 | comment | added | Felipe Voloch | @JT: It's a very old famous paper that remained unpublished for a long time. When the web was young, I took upon myself to make it available online. Now, you can probably just google for it. Anyway, here is the link: ma.utexas.edu/users/voloch/lst.html | |
| Jul 8, 2011 at 23:03 | comment | added | user1594 | +1. Could you indicate which reference is meant by "Lubin-Serre-Tate"? | |
| Jul 8, 2011 at 20:20 | history | answered | Felipe Voloch | CC BY-SA 3.0 |