Dyadic torsion of 2-dimensional hyperelliptic Jacobians

Let $k$ be a field of characteristic $0$, and let ${\alpha }_1$, ${\alpha }_2$, …, ${\alpha }_5$ 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 $J$ be the Jacobian of the hyperelliptic curve defined over $K$ which is given by the equation $y^{2} = \prod_{i = 1}^{5} (x - \alpha_{i})$\mathrm{.}Wedefineatoweroffieldextensions$K = K_{0}’ \subset K_{1}’ \subset K_{2}’ \subset …$ by giving recursive formulas for the generators of each $K_n^{\mathrm{\prime }}$ over $K_{n-1}^{\mathrm{\prime }}$, and let $K_{\infty}’ = \bigcup_{n = 0}^{\infty} K_{n}‘$\mathrm{.}Weshowthat$K_{\infty}’(\mu_{2})$ is the subextension of the field $K(J[2^{\infty}]) := \bigcup_{n = 0}^{\infty} K(E[2^{n}])$correspondingtoacentralorder-$2$Galoissubgroupof$$\mathrm{G}\mathrm{a}\mathrm{l}$(K(J[2^{\infty}]) / K(\mu_{2}))$,andageneratorof$K(J[2^{\infty}])$over$K_{\infty}'(\mu_{2})$ is given.