Set $h(O_E)=0$ and
$$
h(P)=\log H(x(P))
$$
for $P\ne O_E$. The <canonical height of an elliptic curve> is
$$
\widehat h(P)=\frac12\lim_{r\to\infty}4^{-r}h([2^r]P).
$$
The duplication formula is a rational function of degree four in $x$, so the rational-map height estimate in part (a) gives
$$
|h([2]P)-4h(P)|\leq C_E.
$$
Therefore successive terms of $4^{-r}h([2^r]P)$ differ by at most $C_E4^{-r-1}$, and the limit is well defined.

Shifting the limit immediately gives $\widehat h([2]P)=4\widehat h(P)$. The addition formula likewise gives
$$
h(P+Q)+h(P-Q)=2h(P)+2h(Q)+O_E(1).
$$
Apply this to $[2^r]P,[2^r]Q$, divide by $2\cdot4^r$, and pass to the limit to obtain the exact <parallelogram law>
$$
\widehat h(P+Q)+\widehat h(P-Q)=2\widehat h(P)+2\widehat h(Q).
$$
Taking $Q=P$ starts an induction on $|n|$ that yields
$$
\widehat h([n]P)=n^2\widehat h(P)
$$
for every integer $n$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-125/4/b/solution