= Solution
Let $h=\log H$ and set $h_x(P)=h(x(P))$ for $P\ne O$, with $h_x(O)=0$. The given degree-four morphism and part (a) imply that a constant $C$ exists with
$$
|h_x(2P)-4h_x(P)|\leq C
$$
for every $P\in E(\mathbb Q)$. Define
$$
\widehat h(P)=\frac12\lim_{r\to\infty}4^{-r}h_x(2^rP).
$$
To check the limit, put $a_r=4^{-r}h_x(2^rP)$. Then
$$
|a_{r+1}-a_r|\leq C4^{-r-1}.
$$
The geometric series converges, so $(a_r)$ is Cauchy and the limit exists. This is the <canonical height of an elliptic curve>; shifting the sequence by one index immediately gives $\widehat h(2P)=4\widehat h(P)$.
Back to article page
