Timeline for Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 18, 2014 at 4:10 | comment | added | Noam D. Elkies | Yes, they're the only ones. For any elliptic curve $\,E_0$ over a field, the elliptic curves $E$ with $j(E)=j(E_0)$ are precisely the quadratic twists of $E$, unless $E$ has $j$-invariant $0$ or $1728$ which isn't the case here $-$ for us $j(E_0)$ is the function $j \in {\mathbf C}(j)$, which does not equal $0$ or $1728$ in this field. To avoid new punctures we must use quadratic twists by $f(j)$ where $f$ has no zeros or poles outside $\{\infty,0,1728\}$ and there are only four choices including the trivial twist (twists by constants have no effect because ${\bf C}$ is algebraically closed). | |
| Nov 18, 2014 at 4:04 | comment | added | Will Chen | May I ask how you computed the three other elliptic surfaces with the same $j$-invariant? Are these the only other such elliptic surfaces over our twice punctured $j$-line with $j$-invariant $j$? | |
| Nov 17, 2014 at 21:34 | vote | accept | Will Chen | ||
| Nov 17, 2014 at 5:05 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |