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 $n\ge 1$. 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 $\phi (p^n)$. 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 $\phi (p^n)$, and such that the Galois group of $\mathbb{Q}(E[p^n])/\mathbb{Q}$ is as large as possible, i.e., isomorphic to $\mathrm{GL}(2,\mathbb{Z}/p^n\mathbb{Z})$.