Canonical height

Summary

Let $\phi$ be a morphism of degree $d \ge 2$ of projective space over a number field. Its canonical height is the height function adapted to $\phi$: it multiplies by exactly $d$ under $\phi$. It differs from the naive Weil height by a bounded amount, and it vanishes exactly at the preperiodic points of $\phi$. For the multiplication maps on an elliptic curve the same construction gives the Néron–Tate height, a quadratic form on the group of rational points. Canonical heights are a basic tool of arithmetic dynamics.