Computing the canonical height on K3 surfaces

Abstract:

Let $S$ be a surface in $\mathbb {P}^2\times \mathbb {P}^2$ given by the intersection of a (1,1)-form and a (2,2)-form. Then $S$ is a K3 surface with two noncommuting involutions $\sigma ^x$ and $\sigma ^y$. In 1991 the second author constructed two height functions $\hat {h}^+$ and $\hat {h}^-$ which behave canonically with respect to $\sigma ^x$ and $\sigma ^y$, and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights $\sum _v \hat {\lambda }^\pm ( \cdot ,v)$. 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 involutions $\sigma ^x$, $\sigma ^y$, the local heights $\hat {\lambda }^+( \cdot ,v)$, $\hat {\lambda }^-( \cdot ,v)$, and the canonical heights $\hat {h}^+$, $\hat {h}^-$.