An abelian subfield of the dyadic division field of a hyperelliptic Jacobian

Given a field $k$ of characteristic different from $2$ and an integer $d \geq 3$, let $J$ be the Jacobian of the "generic" hyperelliptic curve given by $y^2 = \prod_{i = 1}^d (x - \alpha_i)$, where the $\alpha_i$'s are transcendental and independent over $k$; it is defined over the transcendental extension $K / k$ generated by the symmetric functions of the $\alpha_i$'s. We investigate certain subfields of the field $K_{\infty}$ obtained by adjoining all points of $2$-power order of $J(\bar{K})$. In particular, we explicitly describe the maximal abelian subextension of $K_{\infty} / K(J[2])$ and show that it is contained in $K(J[8])$ (resp. $K(J[16])$) if $g \geq 2$ (resp. if $g = 1$). On the way we obtain an explicit description of the abelian subextension $K(J[4])$, and we describe the action of a particular automorphism in $\mathrm{Gal}(K_{\infty} / K)$ on these subfields.