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 C with CM odd (resp. even) if its endomorphism ring End(E) is an odd (resp. even) order in the imaginary quadratic field End(E)⊗ Q . Suppose that j(E)∈ R and let us consider the set J(R,E) of all j(E) where E is any elliptic curve that enjoys the following properties: . • E is isogenous to E; • j(E)∈ R ; • E has the same parity as E. We prove that the closure of J(R,E) in R is the closed semi-infinite interval (-∞ ,1728] (resp. the whole R ) if E is odd (resp. even). This paper 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.