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

An admissible change of Weierstrass equation of an elliptic curve replaces the $x$-coordinate by a degree-one rational function, so the two logarithmic heights differ by $O(1)$. Dividing this bounded difference by $4^r$ in the defining limit shows that the canonical height is unchanged. It is intrinsic to the elliptic curve and the chosen point.

Replacing $[2^r]$ and $4^r$ by $[3^r]$ and $9^r$ gives the same canonical height. Indeed $h([3]P)=9h(P)+O_E(1)$, and the same telescoping argument constructs a quadratic height differing from the original by a bounded function. A bounded quadratic function is zero, so the two limits agree.

Part (a) givesThus replacing the $x$-coordinate height literally by the $y$-coordinate height multiplies the resulting canonical height by $3/2$. Multiplying the new naive height by $2/3$ restores the original normalization.