Division fields of elliptic curves with minimal ramification

Abstract

Let $E$ be an elliptic curve defined over $\mathbb Q$, let $p$ be a prime number, and let $n\geq 1$. It is well-known that the $p^n$-th division field $\mathbb Q(E[p^n])$ of the elliptic curve $E$ contains all the $p^n$-th roots of unity. It follows that the Galois extension $\mathbb Q(E[p^n])/\mathbb Q$ is ramified above $p$, and the ramification index $e(p,\mathbb Q(E[p^n])/\mathbb Q)$ of any prime $\mathfrak P$ of $\mathbb Q(E[p^n])$ lying above $p$ is divisible by $\varphi(p^n)$. The goal of this article is to construct elliptic curves $E/\mathbb Q$ such that $e(p,\mathbb Q(E[p^n])/\mathbb Q)$ is precisely $\varphi(p^n)$, and such that the Galois group of $\mathbb Q(E[p^n])/\mathbb Q$ is as large as possible, i.e., isomorphic to $\mathrm {GL}(2,\mathbb Z/p^n\mathbb Z)$.