On the Fields of 2-power Torsion of Certain Elliptic Curves

Let μ2∞ denote the group of 2-power roots of unity. The outer pro-2 Galois representation on the projective line minus three points has a kernel whose fixed field, Ω2, is a pro-2 extension of Q (μ2∞), unramified away from 2. The fields of 2-power torsion of elliptic curves defined over Q possessing good reduction away from 2 are also pro-2 extensions of Q (μ2∞), unramified away from 2. In this paper, we show that these fields are contained in Ω2. An analogous result is shown for a certain family of elliptic curves defined over Q (μ2∞).