## Elliptic curves with maximally disjoint division fields

### Volume 175 / 2016

#### Abstract

One of the many interesting algebraic objects associated to a given elliptic curve defined over the rational numbers, $E / \mathbb Q$, is its full-torsion representation $\rho_E:\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\to\operatorname{GL}_2(\hat{\mathbb Z})$. Generalizing this idea, one can create another full-torsion Galois representation $\rho_{(E_1,E_2)}:\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\to(\operatorname{GL}_2(\hat{\mathbb Z}))^2$ associated to a pair $(E_1,E_2)$ of elliptic curves defined over $\mathbb Q$. The goal of this paper is to provide an infinite number of concrete examples of pairs of elliptic curves whose associated full-torsion Galois representation $\rho_{(E_1,E_2)}$ has maximal image. The size of the image is inversely related to the size of the intersection of various division fields defined by $E_1$ and $E_2$. The representation $\rho_{(E_1,E_2)}$ has maximal image when these division fields are maximally disjoint, and most of the paper is devoted to studying these intersections.