Near coincidences and nilpotent division fields

TL;DR

The paper classifies when the division fields $\mathbb{Q}(E[n])$ of elliptic curves $E/\mathbb{Q}$ are nilpotent and introduces near coincidences between division fields. It proves a Gauss–Wantzel analogue for $E:y^{2}=x^{3}-x$ connecting constructibility to $\varphi(n)$ being a power of $2$, and provides a detailed prime- and prime-power level analysis using Galois representations, Cartan subgroups, and modular curves. Conditional on Serre’s Uniformity, it gives a complete composite-level picture, distinguishing CM from non-CM cases and leveraging modular-curve rational points and Magma computations. The results illuminate how division-field nilpotence interacts with the arithmetic of modular curves and the structure of $GL_{2}$ representations, with explicit constraints on which levels can occur and how coincidences arise.

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 $\mathbb{Q}(E[n]) = \mathbb{Q}(E[m],ζ_{n})$. 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 $n \in \{ 3, 5, 6, 7, 15, 21 \}$.

Theorems & Definitions (83)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof
• Proposition 1.5
• Theorem 1.6
• Definition 1.7
• Conjecture 1.8
• Theorem 1.9