= Symmetric-square proof of the naive height parallelogram estimate
{title2=$h_x(P+Q)+h_x(P-Q)=2h_x(P)+2h_x(Q)+O(1)$}
The <biquadratic morphism for elliptic sums and differences> has no common zero, so the local coefficient norms give
$$
h([A:B:C])=2h(x(P))+2h(x(Q))+O(1),
$$
uniformly in $P,Q$. Upper bounds use the fixed coefficients. Lower bounds follow from the absence of common zeros over every <local field>; the constants are $1$ outside finitely many places, where the integral morphism still has no base point after reduction.
For the binary quadratic with roots $u,v\in\mathbb P^1(K)$, the <projective height> of its coefficient vector is $h(u)+h(v)+O(1)$. At nonarchimedean places the coefficient maximum norm of a product of linear <polynomials> is the product of their maximum norms by the <multiplicativity of the nonarchimedean polynomial norm>. At archimedean places the same norms are comparable up to a fixed factor for degree two. This also includes roots at infinity. Combining the comparisons yields $h_x(P+Q)+h_x(P-Q)=2h_x(P)+2h_x(Q)+O(1)$. After dividing the identity for $2^nP,2^nQ$ by $4^n$ and taking limits, it gives the exact <height parallelogram identity> for the <canonical height of an elliptic curve>.
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