Odd and Even Elliptic Curves with Complex Multiplication

We call an order $O$ in a quadratic field $K$ odd (resp. even) if its discriminant is an odd (resp. even) integer. We call an elliptic curve $E$ over the field $\mathbb C$ of complex numbers with $CM$ odd (resp. even) if its endomorphism ring ${\rm End} (E)$ is an odd (resp. even) order in the corresponding imaginary quadratic field. Suppose that $j(E)$ is a real number and let us consider the set $J(R,E)$ of all $j(E')$ where $E'$ is any elliptic curve that enjoys the following properties.
1) $E'$ is isogenous to $E$;
2) $j(E')$ is a real number;
3) $E'$ has the same parity as $E$.
We prove that the closure of $J(R,E)$ in the set $\mathbb R$ of real numbers is the closed semi-infinite interval $(-\infty,1728]$ (resp. the whole $\mathbb R$) if $E$ is odd (resp. even).
This result was inspired by a question of Jean-Louis Colliot-Thélène and Alena Pirutka about the distribution of $j$-invariants of certain elliptic curves of $CM$ type.
The talk will be online.