Timeline for Is it possible to lift a pair of points on an elliptic $\mathbb{F}_{\!q}$-curve to a pair of short points on an elliptic $\mathbb{F}_{\!q}(t)$-curve?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 12, 2023 at 9:49 | comment | added | Dimitri Koshelev | Thank you for your idea. Is it always possible to construct in such a way an isotrivial pencil, that is, all its smooth members are of the same j-invariant? | |
| Jun 11, 2023 at 19:21 | comment | added | Jason Starr | Fine: take the Weirrstrass model, and use a pencil of plane cubics containing the two points that includes the Weirrstrass model as one member of the pencil. The intersection matrix shows the lifts are independent. | |
| Jun 11, 2023 at 11:51 | comment | added | Dimitri Koshelev | I require that the points $P_0(t)$, $P_1(t)$ are independent. In particular, they have to be non-torsion. Hence, your approach does not work. | |
| Jun 11, 2023 at 11:46 | comment | added | Jason Starr | You can just choose $\mathcal{E}$ to be the "constant" family, namely $E\times_{\text{Spec}\ \mathbb{F}_q} \text{Spec}\ \mathbb{F}_q(t)$. Then every $\mathbb{F}_q$-point of $E$ lifts to a "constant" $\mathbb{F}_q(t)$-point of $\mathcal{E}$. | |
| Jun 11, 2023 at 9:44 | comment | added | Dimitri Koshelev | Their coefficients may depend on $q$, but their heights are restricted above by a constant. | |
| Jun 11, 2023 at 9:36 | comment | added | Wojowu | How could $P_0(t)$ etc. ever be independent of $q$ when all the data in your problem depends on it? | |
| Jun 11, 2023 at 9:33 | history | edited | Dimitri Koshelev | CC BY-SA 4.0 |
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| Jun 11, 2023 at 9:27 | history | asked | Dimitri Koshelev | CC BY-SA 4.0 |