Timeline for Distribution of Elliptic Curve Generators over Q
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 3, 2020 at 9:32 | vote | accept | Christopher D. Long | ||
| Dec 3, 2018 at 7:40 | answer | added | François Brunault | timeline score: 6 | |
| Nov 30, 2018 at 13:35 | comment | added | Chris Wuthrich | You are asking if $E(\mathbb{Q})/2E(\mathbb{Q})\to E(\mathbb{R})/2E(\mathbb{R})$ is surjective. Maybe one can say something about the image of the $2$-Selmer group instead; there might be reasons that other local conditions impose the same as the real condition. Instead for the actual rational points, I doubt it is easy to say much. | |
| Nov 30, 2018 at 13:30 | comment | added | Christopher D. Long | Yes, thanks, you are correct that it only makes sense to ask if there is a generator on the non-identity component. | |
| Nov 30, 2018 at 13:13 | comment | added | WhatsUp | Sorry, now I remember that the result for $N$ odd is that this equation doesn't have solutions in positive integers, which is equivalent to saying that all rational points are located on the identity component. | |
| Nov 30, 2018 at 13:10 | comment | added | WhatsUp | Of course, $E(\mathbb{Q})_{tors}$ should a priori be contained in the identity component, for this to make sense. | |
| Nov 30, 2018 at 13:10 | comment | added | François Brunault | Did you try to see where the generators are located for, say, the family of all elliptic curves of rank 1, positive discriminant and trivial torsion? | |
| Nov 30, 2018 at 13:02 | comment | added | WhatsUp | I think there is a result saying that there is no non-trivial rational point for $N$ odd. Also, I don't understand what you mean by "have 2 on identity", etc. It seems to me that it only makes sense to say whether there is a generator on the non-identity component: if $P, Q$ is a set of generators, then $P, P + Q$ also. This is equivalent to asking whether $E(\mathbb{Q})$ is contained in the identity component. | |
| Nov 30, 2018 at 12:35 | history | asked | Christopher D. Long | CC BY-SA 4.0 |