Coincidences of division fields

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let ${\rho }_E:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}(2,\hat{\mathbb{Z}})$ be the adelic representation associated to the natural action of Galois on the torsion points of $E(\overline{\mathbb{Q}})$. By a theorem of Serre, the image of ${\rho }_E$ is open, but the image is always of index at least $2$ in $\mathrm{GL}(2,\hat{\mathbb{Z}})$ due to a certain quadratic entanglement amongst division fields. In this paper, we study other types of abelian entanglements. More concretely, we classify the elliptic curves $E/\mathbb{Q}$, and primes $p$ and $q$ such that $\mathbb{Q}(E[p])\cap \mathbb{Q}({\zeta }_{q^k})$ is non-trivial, and determine the degree of the coincidence. As a consequence, we classify all elliptic curves $E/\mathbb{Q}$ and integers $m,n$ such that the $m$-th and $n$-th division fields coincide, i.e., when $\mathbb{Q}(E[n])=\mathbb{Q}(E[m])$, when the division field is abelian.

28 pages