 Solution

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.