Computing the canonical height on K3 surfaces

Let S be a surface in P 2 × P 2 given by the intersection of a (1,1)-form and a (2,2)-form. Then S is a K3 surface with two noncommuting involutions σ x and σ y . In 1991 the second author constructed two height functions h + and h - which behave canonically with respect to σ x and σ y , and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights Σ υ λ±(.,υ). We discuss how the geometry of the surface S is related to formulas for the local heights, and we give practical algorithms for computing the involution, σ x , σ y ; the local heights λ + (.,υ), λ - (.,υ), and the canonical heights h + , h - .