Timeline for supersingular curve detector
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 17, 2012 at 22:45 | comment | added | stankewicz | Well here's another thing you have to nail down: what do you mean when you say CM elliptic curve? Do you mean over $\mathbf{Q}$ or do you mean over some number field? Because even if you're defined over a number field, the reduction of your elliptic curve can sometimes be defined over $\mathbb{F}_p$. Essentially, Theorem 1.2 of Elkies, Ono and Yang says that for all discriminants $D$ large enough with respect to $p$, all supersingular elliptic curves in char. p can be given thusly in terms of elliptic curves with CM by $\mathcal{O}(D)$. | |
| Nov 17, 2012 at 18:38 | comment | added | Igor Rivin | @Pete L Clark, continued: can you find at least one such? However, this question is too easy, if I understand correctly, if we allow $E$ to have CM, hence the qualification in my question. | |
| Nov 17, 2012 at 18:36 | comment | added | Igor Rivin | @Pete L Clark: I am confused by your comment. Let $E$ be an elliptic curve defined over $\mathbb{Q}$ Its set of reductions will be occasionally (infinitely often, but with low density, as per Elkies) super singular. Presumably, it is very hard to actually compute the set of super singular $p.$ Now, consider the set $\cal{E}$ of all $E$ defined over $\mathbb{Q}$ -- in my primitive way, I am just looking at the set of equations $y^2=x^3 + a x + b,$ with $a, b \in \mathbb{Z}.$ Pick a prime $p \gg 1.$ Then, presumably, there is an $E \in \cal{E}$ such that $E_p$ is super singular. | |
| Nov 17, 2012 at 18:30 | comment | added | Igor Rivin | @stankewicz: are you saying that there is not always a curve I am asking for? I don't disagree, but my impression is that this is a "small prime" phenomenon, such curves always seem to exist over larger primes. | |
| Nov 17, 2012 at 0:47 | comment | added | Pete L. Clark | Igor: every elliptic curve over a finite field is the reduction of a CM elliptic curve over the corresponding unframified extension of Q_p. This is part of the Deuring Lifting Theorem. When the curve is ordinary (= not supersingular) one can choose the lift so that the reduction map on the endomorphism rings is an isomorphism: this is called the canonical lift. Your question seems to need some revision... | |
| Nov 16, 2012 at 22:25 | comment | added | stankewicz | Well in those cases there's only one! I wanted to convey that this was not some weird occurrence, as the paper of Elkies, Ono, and Yang shows. | |
| Nov 16, 2012 at 22:13 | comment | added | Will Sawin | According to wikipedia this is also true for $\mathbb F_2, \mathbb F_3, \mathbb F_5$ and $\mathbb F_7$. | |
| Nov 16, 2012 at 21:24 | comment | added | stankewicz | Over $\mathbb{F}_{11}$ (or the algebraic closure), there are only two supersingular elliptic curves up to isomorphism, one is the reduction of an elliptic curve with j invariant zero and one is the reduction of an elliptic curve with j-invariant 1728. So you must mean something different. | |
| Nov 16, 2012 at 19:58 | comment | added | Igor Rivin | I mean, not a reduction of a CM curve over $\mathbb{Q}.$ Sorry about the shaky terminology, I am just learning about all this. | |
| Nov 16, 2012 at 19:26 | comment | added | Will Sawin | What's the difference between a CM and non-CM supersingular elliptic curve over a finite field? | |
| Nov 16, 2012 at 19:21 | comment | added | David E Speyer | I'm confused. All elliptic curves over $\mathbb{F}_p$ have CM (if CM means endormorphism ring not equal to $\mathbb{Z}$). | |
| Nov 16, 2012 at 18:54 | history | asked | Igor Rivin | CC BY-SA 3.0 |