Elliptic curves with maximally disjoint division fields

Abstract

One of the many interesting algebraic objects associated to a given elliptic curve defined over the rational numbers, E/Q, is its full-torsion representation ρE : Gal(Q/Q)→ GL2(Ẑ). Generalizing this idea, one can create another full-torsion Galois representation, ρ(E1,E2) : Gal(Q/Q)→ ( GL2(Ẑ) )2 associated to a pair (E1, E2) of elliptic curves defined over 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 ρ(E1,E2) has maximal image. The size of the image is inversely related to the size of the intersection of various division fields defined by E1 and E2. The representation ρ(E1,E2) has maximal image when these division fields are maximally disjoint, and most of the paper is devoted to studying these intersections.