Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}\_K(E)\cong \mathcal{O}=\mathcal{O}\_K$. Let $\mathfrak{p}$ be a prime of $K$ such that the map $\mathcal{O}^\times \to(\mathcal{O}/\mathfrak{p})^\times$ is not surjective. With this situation, I tried to prove $E[\mathfrak{p}]\not \subset E(K)$.


My idea is as follows; If $E[\mathfrak{p}]\subset E(K)$, then we can consider $(\mathcal{O}/\mathfrak{p})^\times$ as the set of isomorphisms modulo an equivalence relation `$\sim$' defined by for $f,g\in \mathcal{O}$ we say $f\sim g$ when they coincide on $E[\mathfrak{p}]$. Then it seems contradict to the assumption $\mathcal{O}^\times \to(\mathcal{O}/\mathfrak{p})^\times$ is not surjective. Is it plausible? If it is, how to expand details of proof?



Rational points and torsion points of CM elliptic curve
See Cor. 5.18 of Rubin "Elliptic curves with complex multiplication" in LNM 1716 for a proof (that uses the main theorem of complex multiplication).


