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Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ withJacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} =f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in\mathcal{O}_{K}[x]$ of discriminant $\Delta$ and some element $\lambda \in\mathcal{O}_{K}$. Our first main result says that if there is a prime$\mathfrak{p}$ of $K$ dividing $(f(\lambda))$ but not $(2\Delta)$, then theimage of the natural $2$-adic Galois representation is open in$\mathrm{GSp}(T_{2}(J))$ and contains a certain congruence subgroup of$\mathrm{Sp}(T_{2}(J))$ depending on the maximal power of $\mathfrak{p}$dividing $(f(\lambda))$. We also present and prove a variant of this resultthat applies when $C$ is defined by an equation of the form $y^{2} = f(x)(x -\lambda)(x - \lambda')$ for distinct elements $\lambda, \lambda' \in K$. Wethen show that the hypothesis in the former statement holds for almost all$\lambda \in \mathcal{O}_{K}$ and prove a quantitative form of a uniformboundedness result of Cadoret and Tamagawa.

Boundedness results for 2-adic Galois images associated to hyperelliptic Jacobians