Computing Local p-adic Height Pairings on Hyperelliptic Curves

We describe an algorithm to compute the local component, at a prime above the prime p, of the Coleman–Gross p-adic height pairing on degree 0 divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to evaluate Coleman integrals of meromorphic differentials and present our algorithms as implemented in Sage.