$End_K(E) \cong O_K$($E$ has complex multiplication over $K$ ) implies $K$ is PID?

Question

Let $K$ be a number field, and $E$ be an elliptic curve defined over $K$. $End_K(E) \cong O_K$($E$ has complex multiplication over $K$ ) implies $K$ is PID ?

Reference is also appreciated. Thank you for your help.

Answer 1

Yes, this is the simplest case of class field theory. It is trivial if $K=\Bbb{Q}$. Otherwise $K$ must be an imaginary quadratic field for $O_K$ to be inside the endomorphism ring of a complex torus. Then $Gal(K(j(E))/K) \cong Cl(O_K)$. So $E$ is defined over $K$ means that $j(E)\in K$ which implies that $Cl(O_K)$ is trivial, ie. $O_K$ is a PID.