Computing local p-adic height pairings on hyperelliptic curves

TL;DR

This paper presents an algorithm for computing the local p-adic height pairing on hyperelliptic curves, including new methods for evaluating Coleman integrals, with implementation in Sage.

Contribution

It introduces novel algorithms for calculating local p-adic height pairings and Coleman integrals on hyperelliptic curves, with practical implementation.

Findings

Effective algorithms for local p-adic height computations

New techniques for Coleman integral evaluation

Implementation in Sage software

Abstract

We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on 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.