Timeline for Finding rational points via birational map
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 17, 2021 at 7:46 | comment | added | François Brunault | @monoid911 If you prefer, the places over a singular point $p \in \overline{C}$ are the points of the normalisation of $\overline{C}$ which lie above $p$. Here each singular point $p$ is a node, with the two tangents being defined over $\mathbb{Q}(\sqrt{-3})$ (easy to see at $(0,0)$: the lowest terms of the equation are $x^2+xy+y^2$), so in the normalisation there are two conjugate points above $p$, each defined over $\mathbb{Q}(\sqrt{-3})$. You can think of the normalisation as obtained by separating the tangents. | |
| Feb 16, 2021 at 20:16 | comment | added | monoid911 | Is $\mathbb{Q}(\sqrt{-3})$ a typo? Does "each singular has a place over it" mean that the curve is nonsingular over $\mathbb{Q}_p$ for some prime p? Thank you! | |
| Feb 16, 2021 at 16:38 | comment | added | François Brunault | Sorry, in my last comment I meant the $K$-rational points of $\overline{C}$ minus the two singular points $(0,0)$ and $(-1,-1)$. | |
| Feb 16, 2021 at 13:43 | comment | added | François Brunault | The curve $\overline{C}$ has exactly two singular points, and each of these points has one place over it defined over $\mathbb{Q}(\sqrt{-3})$. So the $K$-rational points of $\overline{C}$ are in bijection with $E_C(K)$ under $\phi$. | |
| Feb 16, 2021 at 13:06 | history | edited | gmvh |
Added top-level tag
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| Feb 16, 2021 at 10:38 | comment | added | Jef | I guess $C$ has a singularity at the origin, so it fails to be smooth. But in any case $\bar{C}$ will only differ from $E_C$ at finitely many points so $\bar{C}$ certainly still has infinitely many $K$-rational points | |
| Feb 16, 2021 at 10:35 | comment | added | Jef | Note that any birational morphism between smooth projective curves uniquely extends to an isomorphism, so if $\bar{C}$ is smooth and $\phi$ is well-defined then the rational points on $E_C$ should correspond exactly to those of $\bar{C}$ under $\phi$ | |
| Feb 16, 2021 at 10:16 | comment | added | Nicolast | Is the pseudo-inverse of $\phi$ defined over $K$ ? | |
| Feb 16, 2021 at 9:52 | review | First posts | |||
| Feb 16, 2021 at 9:53 | |||||
| Feb 16, 2021 at 9:48 | history | asked | monoid911 | CC BY-SA 4.0 |