---
tag: C55ST3TH02
kind: concept
name: Canonical height
msc: [37P30, 11G50, 14G40]
specializes: [F1FF505PWH]
sources:
  - url: https://en.wikipedia.org/wiki/Néron–Tate_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|F1FF505PWH]] 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|BE4X0QG8XW]].

## Explanations

### L1 human

The height of a fraction measures how many digits it needs.
Applying a map like $z \mapsto z^2 + 1$ to a fraction roughly doubles the number of digits, but not exactly.
The canonical height corrects this: it is a slightly adjusted size for which applying the map doubles the size exactly.
Points of canonical height zero are those whose orbit never grows, which are exactly the points that eventually cycle.