Canonical height (reviewed)

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. The canonical height of a rational point $P\in E(\mathbb{Q})$ is computed by writing the $x$-coordinate $x(nP)=A_n(P)/D_n(P)$ as a fraction in lowest terms and setting $$ \hat h(P) = \lim_{n\to\infty} \frac{1}{n^2}\log \max\bigl\{|A_n(P)|,|D_n(P)|\bigr\}. $$ (Note. Some sources define $\hat h$ to be $\frac12$ of this quantity.)

Properties of $\hat h$:

• $\hat h(P)=\log \max\bigl\{|A_1(P)|,|D_1(P)|\bigr\}+O(1)$ as $P$ ranges over $E(\mathbb{Q})$.
• $\hat h(P)\ge0$; and $\hat h(P)=0$ if and only if $P$ is a torsion point.
• $\hat h:E(\mathbb Q)\to\mathbb R$ extends to a positive definite quadratic form on $E(\mathbb{Q})\otimes\mathbb{R}$.

The height pairing on $E$ is the associated bilinear form $\langle P,Q\rangle=\frac{1}{2}\bigl(\hat h(P+Q)-\hat h(P)-\hat h(Q)\bigr)$, which is used to compute the elliptic regulator of $E$. It is a symmetric positive definite bilinear form on $E(\Q)\otimes\R$.

For a number field $K$, the canonical height of $P\in E(K)$ is given by $\hat h(P)=\lim_{n\to\infty} n^{-2}h\bigl(x(nP)\bigr)$, where $h$ is the Weil height.