Timeline for Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 16, 2021 at 19:48 | comment | added | IMT | @Nulhomologous Yes, that's exactly what I mean! | |
| Jan 16, 2021 at 9:17 | comment | added | Nulhomologous | When you say sufficiently large primes, do you mean that there is a bound $b$ such that for all $p>b$ the assertion is true for all curves? | |
| Jan 15, 2021 at 13:57 | comment | added | IMT | @ChrisWuthrich Thanks for pointing that out! I guess I was really interested in what happens for sufficiently large primes $p$, so I've edited the question to reflect that. | |
| Jan 15, 2021 at 13:54 | history | edited | IMT | CC BY-SA 4.0 |
Edited question to restrict to large primes
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| Jan 15, 2021 at 9:33 | comment | added | Chris Wuthrich | Even if $p$ is odd and the elliptic curve $(C,O)$ has good reduction, this may fail. Namely if the reduction $\bar C(\mathbb{F}_p)$ is $2$-torsion, and that can happen for $p\leq 7$, because then $C(\mathbb{Q}_p)/2C(\mathbb{Q}_p) \cong \bar C(\mathbb{F}_p)/2\bar{C}(\mathbb{F}_p)=\bar C(\mathbb{F}_p)$ says that the coset of $P$ modulo $2C(\mathbb{Q}_p)$ is determined by $\bar P$ and hence they have all the same $x$-coordinate. | |
| Jan 15, 2021 at 6:53 | comment | added | Nulhomologous | You can use the Neron model of $E$, which is a group scheme and so it solves your final problem... | |
| Jan 15, 2021 at 2:15 | review | First posts | |||
| Jan 15, 2021 at 6:58 | |||||
| Jan 15, 2021 at 2:14 | history | asked | IMT | CC BY-SA 4.0 |