Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 125 / 5 / b

Let $x:E\to {\mathbb{P}}^1$ be the $x$-coordinate map and define the naive logarithmic height byThe duplication formula induces a degree-four morphism on the $x$-line, so part (a) givesuniformly in $P$. The telescoping sequence $4^{-n}h(2^{n}P)$ is therefore Cauchy. Define the canonical height of an elliptic curve bySumming the geometric error series proves that $|h(P)-\hat{h}(P)|$ is bounded. If another function has this bounded-difference property and scales by four under doubling, evaluating the bounded difference at $2^{n}P$ and dividing by $4^n$ proves uniqueness.

The addition law and part (a) give the approximate parallelogram identityReplace $P,Q$ by $2^{n}P,2^{n}Q$, divide by $4^n$, and let $n\to \infty$. The error disappears, leaving the exact parallelogram lawTogether with $\hat{h}(2P)=4\hat{h}(P)$, induction on $|m|$ yieldsfor every integer $m$.

Let $T$ have order $m>0$. Property (ii) gives $\hat{h}(T)=0$, sinceIt also givesThereforefor every rational torsion point of an elliptic curve. The construction in part (i), quadraticity in part (ii), and this identity establish existence; the bounded-difference and doubling argument in part (i) establishes uniqueness.