Canonical height on an elliptic curve (reviewed)

Let $E$ be an elliptic curve defined over a number field $K$. The canonical height on $E$ is a function $$ \hat{h}: E(K) \to \R_{ {}\ge0} $$ defined on the Mordell-Weil group $E(K)$ which induces a positive definite quadratic form on $E(K)\otimes\R$.

One definition of $\hat{h}(P)$ is $$ \hat h(P)=\lim_{n\to\infty} n^{-2}h\bigl(x(nP)\bigr), $$ where $h(x)$ is the Weil height of $x\in K$. This definition gives the non-normalised height. A normalised height which is invariant under base-change is given by $$ \frac{1}{[K:\Q]} \hat{h}(P). $$

Related to the canonical height is the height pairing $$ \langle-,-\rangle : E(K)\times E(K) \to \R $$ defined by $\langle P,Q\rangle = \frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))$, which is a positive definite quadratic form on $E(K)\otimes\R$, used in defining the regulator of $E/K$.