Revista Matemática Iberoamericana

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Volume 31, Issue 4, 2015, pp. 1311–1332
DOI: 10.4171/RMI/870

Published online: 2015-12-23

Division fields of elliptic curves with minimal ramification

Alvaro Lozano-Robledo[1]

(1) University of Connecticut, Storrs, USA

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 $\mathrm {GL}(2,\mathbb Z/p^n\mathbb Z)$.

Keywords: Elliptic curve, ramification, division field, torsion subgroup

Lozano-Robledo Alvaro: Division fields of elliptic curves with minimal ramification. Rev. Mat. Iberoamericana 31 (2015), 1311-1332. doi: 10.4171/RMI/870