Definition (Height of a point). Let $E∕\mathbb{Q}$ be an elliptic curve, $y^2=x^3+ax+b$.

Define the height

$$\begin{array}{llll}\hfill H:E(\mathbb{Q})& \to {ℝ}_{\ge 1}& \hfill & \\ \hfill P& \mapsto \{\begin{array}{cc}H(x)\hfill & \text{if }P=(x,y)\hfill \\ 1\hfill & \text{if }P=0_E\hfill \end{array}& \hfill & \end{array}$$

Alsdefine logarithmic height

$$\begin{array}{llll}\hfill h:E(\mathbb{Q})& \to {ℝ}_{\ge 0}& \hfill & \\ \hfill P& \mapsto \log H(P)& \hfill & \end{array}$$