COMPOSITE IMAGES OF GALOIS FOR ELLIPTIC CURVES OVER Q AND ENTANGLEMENT FIELDS

Let E be an elliptic curve defined over Q without complex multiplication. For each prime l, there is a representation rho (E,l) : Gal((Q) over bar /Q) -> GL(2)(Z/lZ) that describes the Galois action on the l-torsion points of E. Building on recent work of Rouse-Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over Q with simultaneously non-surjective, composite images of Galois. We also provably determine the rational points on almost all of these curves. Finally, we give an application of our results to the study of entanglement fields.