Odd and Even Elliptic Curves with Complex Multiplication
Yuri G. Zarhin
TL;DR
The paper introduces a parity notion for CM elliptic curves by classifying quadratic orders as odd or even via discriminant and trace, then studies how this parity influences the distribution of CM j-invariants under isogenies. It proves that for CM curves with real j-invariants, the set of real j-values obtained from isogenous curves with the same parity is dense in $(-\infty,1728]$ for odd parity and in $\mathbb{R}$ for even parity, using odd-degree isogenies and a GL$_2(\mathbb{Z}_{(2)})^{+}$-action on the upper half-plane. The work also provides a complete classification and counting of real CM j-invariants with a given odd discriminant, showing exactly $2^{s_D-1}$ such values, where $s_D$ is the number of distinct prime divisors of $D$, and situates them within a precise interval. Together, these results answer questions about the distribution of CM $j$-invariants tied to parity and real structures, connecting arithmetic of quadratic orders to geometric properties of CM elliptic curves.
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 $j(E')$ 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 $(-\infty,1728]$ (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.