Images of 2-adic representations associated to hyperelliptic Jacobians

Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(\alpha_{1}, \alpha_{2}, ... , \alpha_{2g + 1})$, where the $\alpha_{i}$'s are independent and transcendental over $k$, and $g$ is a positive integer. We investigate the image of the $2$-adic Galois action associated to the Jacobian $J$ of the hyperelliptic curve over $K$ given by $y^{2} = \prod_{i = 1}^{2g + 1} (x - \alpha_{i})$. Our main result states that the image of Galois in $\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $\Gamma(2) \lhd \mathrm{Sp}(T_{2}(J))$. As an application, we find generators for the algebraic extension $K(J[4]) / K$ generated by coordinates of the $4$-torsion points of $J$.