The Field of Definition of CM Elliptic Curves

Theorem 3.18   Let $F$ be an elliptic curve over $\mathbb{C}$ with CM by $\O_K$, where $K$ is a quadratic imaginary field. Let $H$ be the Hilbert Class Field of $K$.

1. There is an elliptic curve $E$ defined over $K$ such that $F\cong E_{\mathbb{C}}$.
2. The $\Gal (H/K)$-conjugates of $E$ are representative elements for $\Ell (\O_K)$.
3. If $\sigma \in \Gal (H/K)$ corresponds via Artin reciprocity to $\overline{\mathfrak{a}}\in \Cl (\O_K)$, then
   
   $$E^{\sigma} = \overline{\mathfrak{a}}E.$$
   
   

Theorem 3.18 generalizes in a natural way to the more general situation in which $\O_K$ is replaced by an order $\O_f = \mathbb{Z}+ f\O_K \subset\O_K$. Then the Hilbert class field is replaced by the ray class field $K_f$, which is a finite abelian extension of $H$ that is unramified outside $f$ (see Definition 3.13 above). There is an elliptic curve $E$ defined over $K_f$ whose endomorphism ring is $\O_f$, and the set of $\Gal (K_f/K)$-conjugates of $E$ forms a set of representatives for $\Ell (\O_f)$. Moreover, the group $I(\c_{L/K})/(N\cdot P(\c_{L/K}))$ of Theorem 3.15 acts simply transitively on $\Ell (\O_f)$, and the action of $\Gal (K_f/K)$ on the set of conjugates of $E$ is consistent with the Artin reciprocity map.