CM elliptic curves and vertically entangled 2-adic groups

Abstract

Consider the elliptic curve $E$ given by the Weierstrass equation $y^2=x^3-11x-14$, which has complex multiplication by the order of conductor $2$ inside $\mathbb{Z}[i]$. It was recently observed in a paper of Daniels and Lozano-Robledo that, for each $n\ge 2$, $\mathbb{Q}({\mu }_{2^{n+1}})\subseteq \mathbb{Q}(E[2^n])$. In this note, we prove that this (a priori surprising) ``tower of vertical entanglements'' is actually more a feature than a bug: it holds for any elliptic curve $E$ over $\mathbb{Q}$ with complex multiplication by any order of even discriminant.