Canonical height of an elliptic curve ($\hat{h}$)

For an elliptic curve over a number field, fix the absolute logarithmic $x$-height $h_x(P)=h([x(P):1])$, with $h_x(O)=0$. The normalization used here isThe factor $1/2$ makes this the Néron–Tate height associated with the degree-one divisor $(O)$. Some sources omit $1/2$ and obtain twice this function; formulas and the regulator of an elliptic curve must use a consistent convention.^[ref]

The estimate $h_x(2P)=4h_x(P)+O(1)$ makes successive terms differ by a summable geometric series, proving convergence and $\hat{h}(P)=h_x(P)/2+O(1)$. The resulting quadratic form satisfies $\hat{h}([m]P)=m^2\hat{h}(P)$ and the height parallelogram identity. It is nonnegative and vanishes exactly on torsion points of an elliptic curve; bounded-height rational points over a fixed number field form a finite set. For an isogeny $\varphi :E\to E^{\prime }$, the corresponding normalized heights satisfy ${\hat{h}}_{E^{\prime }}(\varphi P)=\mathrm{deg}(\varphi ){\hat{h}}_E(P)$.

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.

The canonical height pairing is $⟨P,Q⟩=(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))/2$. The height parallelogram identity makes it bilinear. It descends through torsion and is positive definite on the free part of the Mordell-Weil group. The Cauchy-Schwarz inequality gives $\hat{h}(P-Q)\le 2\hat{h}(P)+2\hat{h}(Q)$, useful for height descent.

The Gram matrix of the canonical height pairing of rational points gives a robust independence certificate. For two points, set $a=\hat{h}(P)$, $b=\hat{h}(Q)$ and $c=(\hat{h}(P+Q)-a-b)/2$. If bounds from rounded measurements imply $a>0$ and $ab-c^2>0$, the matrix is positive-definite. Any integer relation $rP+sQ=O$ would give $a{r}^2+2crs+b{s}^2=0$, forcing $r=s=0$. Thus the points generate a free abelian group of rank two. Checking an interval lower bound for the determinant, rather than just its approximate central value, makes the certificate insensitive to rounding. It certifies independence, not that the points are a basis of the entire Mordell-Weil group.

If an elliptic curve over $\mathbb{Q}$ has Mordell-Weil group of rank $r$, then its canonical height of an elliptic curve is a positive-definite quadratic form on the free lattice. Consequently

The regulator is the determinant of the canonical-height pairing on a basis of the free part of the Mordell-Weil group. A unimodular basis change leaves it invariant.