Let $E/\Rats$ be an elliptic curve. It is often useful to have a notion of height of a point, in order to talk about the arithmetic complexity of a point $P$ in $E(\Rats)$ For this, one defines height functions. For example, in $\Rats$ one can define a height by $$H(p/q)=max(|p|,|q|).$$ Following the example of $\Rats$ one may define a height on $E/\Rats$ by $$h_x(P)=\begin{cases} \log H(x(P)) & \text{if } P\neq O\\ 0 & \text{if } P=O. \end{cases} $$ In fact, given any even function $f:E(\Rats)\to \Reals$ on $E(\Rats)$ (i.e. $f(P)=f(-P)$ for any $P\in E(\Rats)$ one can define a height by: $$h_f(P)=\log H(f(P)).$$ However, one can refine this definition so that the height function satisfies some very nice properties (see below).

The fact that the definition does not depend on the choice of even function $f$ is due to J. Tate. In particular, one can simply choose $f$ to be the $x$ function, whose degree is $2$ The canonical height satisfies the following properties: