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 C of complex numbers with CM odd (resp. even) if its endomorphism ring 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 R of real numbers 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-Th\'el\`ene and Alena Pirutka about the distribution of j-invariants of certain elliptic curves of CM type.