Elliptic curves with maximally disjoint division fields

Abstract

One of the many interesting algebraic objects associated to a given rational elliptic curve, $E$, is its full-torsion representation ${\rho }_E:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to {\mathrm{GL}}_2(\hat{\mathbf{Z}})$. Generalizing this idea, one can create another full-torsion Galois representation, ${\rho }_{(E_1,E_2)}:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to {\left({\mathrm{GL}}_2(\hat{\mathbf{Z}})\right)}^2$ associated to a pair $(E_1,E_2)$ of rational elliptic curves. 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.