CM elliptic curves and vertically entangled 2-adic groups

TL;DR

This paper investigates the structure of 2-adic Galois representations of elliptic curves with complex multiplication, revealing a consistent pattern of vertical entanglements across such curves.

Contribution

It generalizes the phenomenon of vertical entanglements in 2-adic Galois groups from specific CM elliptic curves to all CM elliptic curves with even discriminant.

Findings

The tower of field extensions $Q(mu_{2^{n+1}}) subseteq Q(E[2^n])$ holds for all CM elliptic curves with even discriminant.

Vertical entanglements are a common feature in the 2-adic Galois representations of these elliptic curves.

The result extends previous observations from particular cases to a broad class of CM elliptic curves.

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 $n \geq 2$, $\mathbb{Q}(\mu_{2^{n+1}}) \subseteq \mathbb{Q}(E[2^n])$. 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.