 Symmetric-square proof of the naive height parallelogram estimate

The biquadratic morphism for elliptic sums and differences has no common zero, so the local coefficient norms giveuniformly 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 $2^{n}P,2^{n}Q$ by $4^n$ and taking limits, it gives the exact height parallelogram identity for the canonical height of an elliptic curve.