The Set of CM Elliptic Curves with Given CM

Definition 3.6 (Fractional Ideal)   A fractional ideal $\mathfrak{a}$ of a number field $K$ is an $\O_K$-submodule of $K$ that is isomorphic to $\mathbb{Z}^{[K:\mathbb{Q}]}$ as an abelian group. In particular, $\mathfrak{a}$ is nonzero.

If $\mathfrak{a}$ is a fractional ideal, the inverse $\mathfrak{a}^{-1}$ of $\mathfrak{a}$, which is the set of $x \in K$ such that $x\mathfrak{a}\subset \O_K$, is also a fractional ideal. Moreover, $\mathfrak{a}\mathfrak{a}^{-1} = \O_K$.

Fix a quadratic imaginary field $K$. Let $\Ell (\O_K)$ be the set of $\mathbb{C}$-isomorphism classes of elliptic curves $E/\mathbb{C}$ with $\End (E)\cong \O_K$. By the above results we may also view $\Ell (\O_K)$ as the set of lattices $\Lambda$ with $\End (E_{\Lambda}) \cong \O_K$.

If $\mathfrak{a}$ is a fractional $\O_K$ ideal, then $\mathfrak{a}\subset K\subset \mathbb{C}$ is a lattice in $\mathbb{C}$. For the elliptic curve $E_{\mathfrak{a}}$ we have

$$\End (E_\mathfrak{a}) = \O_K,$$

because $\mathfrak{a}$ is an $\O_K$-module by definition. Since rescaling a lattice produces an isomorphic elliptic curve, for any nonzero $c\in K$ the fractional ideals $\mathfrak{a}$ and $c\mathfrak{a}$ define the same elements of $\Ell (\O_K)$.

The class group $\Cl (\O_K)$ is the group of fractional ideals modulo principal fractional ideals. If $\mathfrak{a}$ is a fractional $\O_K$ ideal, denote by $\overline{\mathfrak{a}}$ its ideal class in the class group $\Cl (\O_K)$ of $K$. We have a natural map

$$\Cl (\O_K) \to \Ell (\O_K),$$

which sends $\overline{\mathfrak{a}}$ to $E_{\mathfrak{a}}$.

Theorem 3.7   Fix a quadratic imaginary field $K$, and let $\Lambda$ be a lattice in $\mathbb{C}$ such that $E_{\Lambda} \in \Ell (\O_K)$. Let $\mathfrak{a}$ and $b$ be nonzero fractional $\O_K$-ideals. Then

1. $\mathfrak{a}\Lambda$ is a lattice in $\mathbb{C}$,
2. We have $\End (E_{\mathfrak{a}\Lambda}) \cong \O_K$.
3. We have $E_{\mathfrak{a}\Lambda}\cong E_{\b\Lambda}$ if and only if $\overline{\mathfrak{a}}= \overline{\mathfrak{b}}$.

Thus there is a well-defined action of $\Cl (\O_K)$ on $\Ell (\O_K)$ given by

$$\overline{\mathfrak{a}}E_{\Lambda} = E_{\overline{\mathfrak{a}}^{-1} \Lambda}.$$

Theorem 3.8   The action of $\Cl (\O_K)$ on $\Ell (\O_K)$ is simply transitive.

Example 3.9   Let $K=\mathbb{Q}(\sqrt{-23})$. Then the class number $h_K$ is $3$. An elliptic curve with CM by $\O_K$ is $\mathbb{C}/(\mathbb{Z}+(1+\sqrt{-23})/2 \mathbb{Z})$, and one can obtain the other two elements of $\Ell (\O_K)$ by multiplying the lattice $\mathbb{Z}+(1+\sqrt{-23})/2 \mathbb{Z}$ by two representative ideal classes for $\Cl (\O_K)$.