Entanglement in the family of division fields of elliptic curves with complex multiplication

TL;DR

This paper investigates the entanglement properties of division fields of CM elliptic curves over number fields, establishing conditions for linear disjointness and describing entanglement in specific cases.

Contribution

It proves linear disjointness of division fields after removing a finite subfamily and characterizes entanglement conditions for CM elliptic curves over their CM field.

Findings

Family of division fields becomes linearly disjoint after removing finite subfamily

Necessary condition for entanglement over the base field

Explicit description of entanglement when the curve is a base change from Q

Abstract

For every elliptic curve $E$ which has complex multiplication (CM) and is defined over a number field $F$ containing the CM field $K$, we prove that the family of $p^{\infty}$-division fields of $E$, with $p \in \mathbb{N}$ prime, becomes linearly disjoint over $F$ after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over $F$, which is always met when $F = K$. In this case, and under the further assumption that the elliptic curve $E$ is obtained as a base-change from $\mathbb{Q}$, we describe in detail the entanglement in the family of division fields of $E$.