Near coincidences and nilpotent division fields

Harris B. Daniels and Jeremy Rouse

Vol. 342 (2026), No. 1, 79–113

DOI: 10.2140/pjm.2026.342.79

Abstract

Let $E∕\mathbb{Q}$ be an elliptic curve. We say that $E$ has a near coincidence of level $(n,m)$ if $m|n$ 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 $\mathrm{Gal}<!--FUNCTION\; APPLICATION-->(\mathbb{Q}(E[n])∕\mathbb{Q})$ is a nilpotent group. Along the way we prove a Gauss–Wantzel analog for the elliptic curve $E:y^2=x^3-x$, showing that $\mathbb{Q}(E[n])∕\mathbb{Q}$ is constructible if and only if $\phi (n)$ 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 $\mathrm{Gal}<!--FUNCTION\; APPLICATION-->(\mathbb{Q}(E[n])∕\mathbb{Q})$ nilpotent implies that $n$ is a power of $2$ or $n\in \{3,5,6,7,15,21\}$.

Keywords

elliptic curve, division field, Galois group, modular curve

Mathematical Subject Classification

Primary: 11G05

Secondary: 11R32, 14H52

Milestones

Received: 22 December 2024

Revised: 9 December 2025

Accepted: 2 February 2026

Published: 23 March 2026