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

Let K be a number field, and let C be a hyperelliptic curve over K with Jacobian J. Suppose that C is defined by an equation of the form y2=f(x)(x-lambda) for some irreducible monic polynomial f is an element of OK[x] of discriminant Delta and some element lambda is an element of OK. Our first main result says that if there is a prime p of K dividing (f(lambda)) but not (2 Delta), then the image of the natural 2-adic Galois representation is open in GSp(T2(J)) and contains a certain congruence subgroup of Sp(T2(J)) depending on the maximal power of p dividing (f(lambda)). We also present and prove a variant of this result that applies when C is defined by an equation of the form y2=f(x)(x-lambda)(x-lambda ') for distinct elements lambda,lambda 'is an element of K. We then show that the hypothesis in the former statement holds for almost all lambda is an element of OK and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.