Indices of hyperelliptic curves over p-adic fields

Let k be a p-adic field of odd residue characteristic and let C be a hyperelliptic (or elliptic) curve defined by the affine equation Y-2 = h(X). We discuss the index of C if h(X) = epsilon f(X), where epsilon is either a non-square unit or a uniformising element in O-k and f(X) a monic, irreducible polynomial with integral coefficients. If a root theta of f generates an extension k(theta) with ramification index a power of 2, we completely determine the index of C in terms of data associated to theta. Theorem 3.11 summarizes our results and provides an algorithm to calculate the index for such curves C.