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

Since $\hat{h}-h$ is bounded, a bound on $\hat{h}(P)$ bounds the projective height of $x(P)$. The Northcott theorem gives only finitely many possible rational $x$-coordinates, and each has at most two points above it. Hence

By the Mordell-Weil theorem, $E(\mathbb{Q})\cong {\mathbb{Z}}^r\oplus E(\mathbb{Q}{)}_{\mathrm{tors}}$. The canonical height is a positive-definite quadratic form on the lattice ${\mathbb{Z}}^r$, so canonical-height lattice-point growth givesIf $r=0$, this count is bounded; if $r=1$, it is $O(\sqrt{B})$. Consequently