5
$\begingroup$

Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\overline{C}/\mathbb{F}_p$ denote the mod-$p$ reduction of $C$, and for any $P \in C(\mathbb{Q}_p)$, let $\overline{P} \in \overline{C}(\mathbb{F}_p)$ denote the mod-$p$ reduction of $P$.

Suppose that $C(\mathbb{Q}_p) \neq \varnothing$, and fix $O \in C(\mathbb{Q}_p)$. Let $E$ be the Jacobian of $C$, and let $\iota \colon C(\mathbb{Q}_p) \to E(\mathbb{Q}_p)$ be the map sending $P \mapsto [P - O]$.

Question: Suppose $p \gg 1$. Let $P \in C(\mathbb{Q}_p)$ be integral, meaning that $x(\overline{P}) \neq \infty$. Does there exist $Q \in C(\mathbb{Q}_p)$ such that $\iota(P) + 2 \cdot \iota(Q) = \iota(R)$, where $R \in C(\mathbb{Q}_p)$ is integral and $x(\overline{P}) \neq x(\overline{R})$?

Partial Answer: Suppose that $C$ has good reduction modulo $p$ (i.e., suppose that $p$ does not divide the discriminant of $C$), and let $\overline{E}/\mathbb{F}_p$ denote the mod-$p$ reduction of $E$. Then $\#\overline{C}(\mathbb{F}_p) \gg p$ by the Hasse bound, so the set of points of the form $\iota(\overline{P}) + 2 \cdot \iota(\overline{Q}) \in \overline{E}(\mathbb{F}_p)$ has size $\gg p$. But the set of points $\overline{R} \in \overline{C}(\mathbb{F}_p)$ such that $x(\overline{R}) \in \{\infty, x(\overline{P})\}$ has size $\ll 1$, so the answer to the question in this case is yes if $p$ is sufficiently large. I'm not sure how to make the above argument work in the case where $E$ does not have good reduction modulo $p$, because I can't talk about the group $\overline{E}(\mathbb{F}_p)$.

Edits: I've edited the question to include the assumption $p \gg 1$, so as to avoid the issue mentioned by Chris Wuthrich, which is that $\overline{C}(\mathbb{F}_p)$ can be $2$-torsion.

Improve this question
$\endgroup$
5
  • $\begingroup$ You can use the Neron model of $E$, which is a group scheme and so it solves your final problem... $\endgroup$
    – Nulhomologous
    Commented Jan 15, 2021 at 6:53
  • 2
    $\begingroup$ 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. $\endgroup$
    – Chris Wuthrich
    Commented Jan 15, 2021 at 9:33
  • $\begingroup$ @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. $\endgroup$
    – IMT
    Commented Jan 15, 2021 at 13:57
  • $\begingroup$ 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? $\endgroup$
    – Nulhomologous
    Commented Jan 16, 2021 at 9:17
  • $\begingroup$ @Nulhomologous Yes, that's exactly what I mean! $\endgroup$
    – IMT
    Commented Jan 16, 2021 at 19:48

0

Reset to default

You must log in to answer this question.