Timeline for Supersingular curves over $\mathbb{F}_q$ and the splitting of $p$
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Jul 29, 2021 at 19:35 | history | bounty ended | Rdrr | ||
| S Jul 29, 2021 at 19:35 | history | notice removed | Rdrr | ||
| Jul 29, 2021 at 19:35 | vote | accept | Rdrr | ||
| Jul 29, 2021 at 13:37 | vote | accept | Rdrr | ||
| Jul 29, 2021 at 19:35 | |||||
| Jul 28, 2021 at 1:06 | answer | added | Bjorn Poonen | timeline score: 5 | |
| Jul 24, 2021 at 14:16 | answer | added | Yuri Zarhin | timeline score: 4 | |
| Jul 23, 2021 at 21:23 | comment | added | Will Sawin | The way I see how to do it is to observe that the endomorphism algebra is a quaternion algebra, and (by the $\ell$-adic Tate module) split at each prime $\ell$ not $p$. If $\mathbb Q(\alpha)$ is split at $p$ then it is split everywhere, hence a matrix algebra, thus contains nilpotents, which is absurd. Is this the argument of Lang you mention? I don't see a better way. | |
| Jul 23, 2021 at 19:50 | comment | added | Rdrr | That's exactly where I get stuck; I would like to know what extra information could finish off the proof. | |
| Jul 23, 2021 at 19:15 | comment | added | Will Sawin | I don't think it's possible to finish the argument from where you are because you need to rule out polynomials like $x^2 - \sqrt{q} x + q$ for $p$ congruent to $1$ mod $3$ and $q$ a square, but roots $\alpha$ of these polynomials satisfy the condition $\alpha^N \in \mathbb Z$. So you need more information about elliptic curves. | |
| Jul 23, 2021 at 18:52 | history | edited | Rdrr | CC BY-SA 4.0 |
added 1073 characters in body
|
| S Jul 23, 2021 at 18:36 | history | bounty started | Rdrr | ||
| S Jul 23, 2021 at 18:36 | history | notice added | Rdrr | Authoritative reference needed | |
| Jul 21, 2021 at 19:26 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
|
| Jul 21, 2021 at 18:59 | history | edited | Rdrr | CC BY-SA 4.0 |
added 227 characters in body
|
| Jul 21, 2021 at 18:01 | history | asked | Rdrr | CC BY-SA 4.0 |