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:

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

Here $h$ denotes the naive height.

Facts

Identities

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

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