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−λ) for some irreducible monic polynomial f∈OK[x] of discriminant Δ and some element λ∈OK. Our first main result says that if there is a prime p of K dividing (f(λ)) but not (2Δ), 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(λ)). 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−λ)(x−λ′) for distinct elements λ,λ′∈K. We then show that the hypothesis in the former statement holds for almost all λ∈OK and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.