Odd and Even Elliptic Curves with Complex Multiplication

TL;DR

This paper investigates the distribution of j-invariants of CM elliptic curves with odd or even endomorphism rings, showing their limits form specific intervals depending on parity, inspired by questions on CM elliptic curves.

Contribution

It characterizes the closure of the set of real j-invariants of isogenous CM elliptic curves with matching parity, revealing their distribution over specific real intervals.

Findings

Closure of j-invariants set is $(- abla,1728]$ for odd CM elliptic curves.

Closure of j-invariants set is the entire real line for even CM elliptic curves.

Distribution depends on the parity of the endomorphism ring in 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-Th\'el\`ene and Alena Pirutka about the distribution of $j$-invariants of certain elliptic curves of CM type.