Height parallelogram identity ($\hat{h}(P+Q)+\hat{h}(P-Q)=2\hat{h}(P)+2\hat{h}(Q)$)

For naive logarithmic $x$-height on an elliptic curve, $h_x(P+Q)+h_x(P-Q)=2h_x(P)+2h_x(Q)+O(1)$ uniformly. The unordered pair of sum and difference on the $x$-line defines a bidegree-$(2,2)$ morphism to ${\mathrm{Sym}}^2{\mathbb{P}}^1$, yielding the estimate. Applying it to $2^{n}P,2^{n}Q$ and taking the defining limit gives the exact identity for the canonical height of an elliptic curve.

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.

For a nonsingular Short Weierstrass form $y^2=x^3+ax+b$ over a number field, the unordered pair of coordinates $x(P+Q),x(P-Q)$ depends only on $x(P),x(Q)$. As a binary quadratic's roots, that pair is represented in ${\mathbb{P}}^2$ by the coefficientswith polynomial $C{T}^2-BT+A$. Homogenizing gives three forms of bidegree $(2,2)$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. They have no common zero. Off the diagonal $C\ne 0$; on the finite diagonal $x_1=x_2=t$, $B=4f(t)$ and $A=f^{\prime }(t{)}^2-8tf(t)$ for $f(t)=t^3+at+b$, so simultaneous vanishing would make a multiple root of $f$. At $(\infty ,\infty )$, the homogeneous $A$ is nonzero. Therefore they define a global morphism, also when a sum or difference is the identity.