 Solution

For $x(P)=A/B^2$ in lowest terms, take the logarithmic naive height $h(P)=\log \max \{|A|,B^2\}$. Its required properties areandwith constants depending only on the curve. Define the canonical height of an elliptic curve byThe first bounded-error relation makes this a convergent telescoping correction to $h(P)$. Apply the second relation to $2^{r}P,2^{r}Q$, divide by $4^r$, and let $r\to \infty$ to obtainAlso $\hat{h}(2P)=4\hat{h}(P)$, and the parallelogram identity then gives $\hat{h}(nP)=n^2\hat{h}(P)$ for every integer $n$. Thus $\hat{h}$ is a quadratic form.