Canonical height

Definition

Let $K$ be a number field and $E$ be an elliptic curve. The canonical height $\hat{h}$ is a map from the points of $E(K)$ to elements of $K$ (or the completion of $K$) defined as follows:

$\hat{h}(Q)=\underset{n\to \mathrm{\infty }}{lim}{4}^{-n}h(2^{n}Q)$

Here $h$ denotes the naive height.

Facts

Identities

The canonical height satisfies the following identities, that indicate that it is something quadratic in nature:

• $\hat{h}(P+Q)+\hat{h}(P-Q)=2(\hat{h}(P)+\hat{h}(Q))\mathrm{\forall }P,Q\in E(K)$
• $\hat{h}(Q)=0$ for all torsion points $Q\in E(K)$ (that is, for all points of finite order)