The maximal abelian extension contained in a division field of an elliptic curve over $\mathbb{Q}$ with complex multiplication

TL;DR

This paper investigates the abelian extensions within division fields of CM elliptic curves over rationals, providing bounds on Galois group commutators and classifying maximal abelian subextensions.

Contribution

It introduces bounds on the commutator subgroups of Galois groups and classifies maximal abelian extensions in division fields of CM elliptic curves over .

Findings

Bounded the commutator subgroups of Galois groups for division fields.

Classified the maximal abelian extensions contained in these division fields.

Extended previous results to higher powers of primes in division fields.

Abstract

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$, where $j_{K,f}=j(E)$. It has been shown by the author and Lozano-Robledo that $\operatorname{Gal}(\mathbb{Q}(j_{K,f},E[N])/\mathbb{Q}(j_{K,f}))$ is only abelian for $N=2,3$, and $4$. Let $p$ be a prime and let $n\geq 1$ be an integer. In this article, we bound the commutator subgroups of $\operatorname{Gal}(\mathbb{Q}(E[p^n])/\mathbb{Q})$ and classify the maximal abelian extensions contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$.