Near coincidences and nilpotent division fields

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 $\mathbb{Q}(E[n])=\mathbb{Q}(E[m],\hat{I}{\P }_n)$. We classify near coincidences of prime power level and use this result to give a classification of values of $n$ for which $\mathrm{Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ 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 $\ddot{I}†(n)$ 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 $\mathrm{Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ nilpotent implies that $n$ is a power of $2$ or $n\in \{3,5,6,7,15,21\}$.

29 pages