Timeline for Prime splitting in the division field of an elliptic curve
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 16 at 13:58 | comment | added | Jeff H | @ChrisWuthrich You're right, I accidentally wrote down the Duke-Toth matrix for when a_p=2 rather than -2, but it doesn't affect the rest of my question -- in particular, it was not the source of my error. But thanks for catching that. | |
| Feb 16 at 13:56 | comment | added | Jeff H | @re'emwaxman Yes, that's exactly the point I was somehow missing, but your comment made my false belief clear to me. Thanks so much again! | |
| Feb 16 at 11:54 | comment | added | re'em waxman | The thing needed to be understood is that if $\mathfrak{p}$ and $\mathfrak{p}'$ are two primes above $p$ then $\text{Frob}_{\mathfrak{p}}$ and $\text{Frob}_{\mathfrak{p}'}$ are conjugate in $G$ but might be the same even if the primes are distinct. | |
| Feb 16 at 11:15 | comment | added | Chris Wuthrich | Also your example doesn't have trace equal to $a_p=-2$. | |
| Feb 15 at 21:37 | comment | added | Jeff H | @re'emwaxman : Thanks -- I figured my mistake had to be at that step. I'm still a bit confused about it, but your extreme example of an abelian extension is extremely helpful for convincing me that this is where I've goofed. | |
| Feb 15 at 21:32 | comment | added | GH from MO | Please use a high-level tag like "nt.number-theory". I added this tag now. | |
| Feb 15 at 21:31 | history | edited | GH from MO | CC BY-SA 4.0 |
edited body
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| Feb 15 at 21:30 | comment | added | re'em waxman | I think your number theory is wrong. There is no correspondence between elements in $\mathcal{C}$ to primes above $p$, for example take $K/\mathbb{Q}$ abelian and any prime $p$ which does stay prime in $K$, then $|\mathcal{C}|=1$ and $g>1$. | |
| Feb 15 at 19:51 | history | asked | Jeff H | CC BY-SA 4.0 |