Wild Galois representations: a family of hyperelliptic curves with large inertia image

In this work we generalise the main result of [1] to the family of hyperelliptic curves with potentially good reduction over a p-adic field which have genus g=(p-1)/2 and the largest possible image of inertia under the ℓ-adic Galois representation associated to its Jacobian. We will prove that this Galois representation factors as the tensor product of an unramified character and an irreducible representation of a finite group, which can be either equal to the inertia image (in which case the representation is easily determined) or a C2-extension of it. In this second case, there are two suitable representations and we will describe the Galois action explicitly in order to determine the correct one.