Odd and Even Elliptic Curves with Complex Multiplication

Abstract

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 $j(E^{\prime })$ where $E^{\prime }$ is any elliptic curve that enjoys the following properties. 1) $E^{\prime }$ is isogenous to $E$; 2) $j(E^{\prime })$ is a real number; 3) $E^{\prime }$ 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 $(-\infty ,1728]$ (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.

27 pages