Near coincidences and nilpotent division fields

TL;DR

This paper classifies near coincidences of elliptic curve division fields and characterizes when these fields have nilpotent Galois groups, revealing connections to constructibility and specific prime power levels.

Contribution

It provides a classification of near coincidences of prime power level and characterizes when elliptic curve division fields have nilpotent Galois groups, including a Gauss-Wantzel analog.

Findings

Classified near coincidences of prime power level.

Identified when division fields are constructible based on Euler's totient.

Determined possible n for nilpotent Galois groups under certain assumptions.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve. We say that $E$ has a near coincidence of level $(n,m)$ if $m \mid n$ and $\mathbb{Q}(E[n]) = \mathbb{Q}(E[m],\zeta_{n})$. We classify near coincidences of prime power level and use this result to give a classification of values of $n$ for which ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ is a nilpotent group. Along the way we prove a Gauss-Wantzel analog for the elliptic curve $E\colon y^2 = x^3-x$, showing that $\mathbb{Q}(E[n])/\mathbb{Q}$ is constructible if and only if $\varphi(n)$ is a power of 2. Assuming that there are no non-CM rational points on the modular curves $X_{ns}^{+}(p)$ for primes $p > 11$, we show that ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ nilpotent implies that $n$ is a power of $2$ or $n \in \{ 3, 5, 6, 7, 15, 21 \}$.