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 formy 2 = f ( x ) ( x − λ )for some irreducible monic polynomial f ∈ O K [ x ]of discriminant Δ and some element λ ∈ O K . 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 ( T 2 ( J ) ) and contains a certain congruence subgroup of Sp ( T 2 ( 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 formy 2 = 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 ∈ O Kand prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.