Dyadic Torsion of Elliptic Curves

Abstract

Let $k$ be a field of characteristic $0$, and let $\alpha_{1}$, $\alpha_{2}$, and $\alpha_{3}$ be algebraically independent and transcendental over $k$. Let $K$ be the transcendental extension of $k$ obtained by adjoining the elementary symmetric functions of the $\alpha_{i}$'s. Let $E$ be the elliptic curve defined over $K$ which is given by the equation $y^{2} = (x - \alpha_{1})(x - \alpha_{2})(x - \alpha_{3})$. We define a tower of field extensions $K = K_{0}' \subset K_{1}' \subset K_{2}' \subset ...$ by giving recursive formulas for the generators of each $K_{n}'$ over $K_{n - 1}'$. We show that $K_{\infty}'$ is a certain central subextension of the field $K(E[2^{\infty}]) := \bigcup_{n = 0}^{\infty} K(E[2^{n}])$, and a generator of $K(E[2^{\infty}])$ over $K_{\infty}'(\mu_{2})$ is given. Moreover, if we assume that $k$ contains all $2$-power roots of unity, for each $n$, we show that $K(E[2^{n}])$ contains $K_{n}'$ and is contained in a certain quadratic extension of $K_{n + 1}'$.