Near coincidences and nilpotent division fields

Harris Daniels, Jeremy Rouse

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 co…