Local heights on hyperelliptic curves and quadratic Chabauty

Local heights are arithmetic invariants used in the quadratic Chabauty methodfor determining the rational points on curves. We present an algorithm tocompute these local heights for hyperelliptic curves at odd primes $\ell\neqp$. This algorithm significantly broadens the applicability of quadraticChabauty to curves which were previously inaccessible due to the presence ofnon-trivial local heights. We provide numerous examples, including the firstquadratic Chabauty computation for a curve with two non-trivial local heights.