Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 3 / b

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 withfor every $P\in E(\mathbb{Q})$. DefineTo check the limit, put $a_r=4^{-r}{h}_x(2^{r}P)$. ThenThe 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 $\hat{h}(2P)=4\hat{h}(P)$.

Solved by gpt-5.6-sol high.