Let $p$ be a fixed prime. Say for simplicity $p>5$. As we vary over all elliptic curves $E/\mathbb{Q}$ of height $< X$, can one (expect to) say anything about what proportion of elliptic curves might satisfy the property: $p = \# \tilde{E}(\mathbb{F}_p)$? Here $\tilde{E}$ is the reduced curve $\mod p$ and $\mathbb{F}_p$ is the finite field with $p$ elements.
$\begingroup$
$\endgroup$
2
-
3$\begingroup$ It should a positive proportion. More generally, for any elliptic curve $E_p$ over $\mathbb{F}_p$, a positive proportion of elliptic curves $E$ realise $E_p$ modulo $p$ since you can arrange this by imposing congruence conditions on the coefficients of $E$. Do you want to know some formula for the exact proportion? $\endgroup$– Daniel LoughranJan 26, 2021 at 14:51
-
$\begingroup$ It would be helpful to know some exact formulae if available. $\endgroup$– debanjanaJan 26, 2021 at 14:57
Add a comment
|