On the fields of $2$-power torsion of certain elliptic curves

Let $\mu_{2^\infty}$ 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, $\Omega_2$, is a pro-$2$ extension of $\Q \left( \mu_{2^\infty}\! \right)$, 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 $\Q \left( \mu_{2^\infty}\! \right)$, unramified away from $2$. In this paper, we show that these fields are contained in $\Omega_2$. An analogous result is shown for a certain family of elliptic curves defined over $\Q \left( \mu_{2^\infty}\! \right)$.

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Published 1 January 2004