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:

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

Here $$h$$ denotes the naive height.

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)