= Canonical height of an elliptic curve
{title2=$\widehat h$}
= Canonical height
{synonym}
= Normalized height on an elliptic curve
{synonym}
= Normalized height
{synonym}
This is the elliptic-curve specialization of the <Néron–Tate height>. For an associated logarithmic projective height $h$, one normalization is $\widehat h(P)=\lim_{n o\infty}4^{-n}h([2^n]P)$.
The canonical height is
$$
\widehat h(P)=\frac12\lim_{r\to\infty}4^{-r}h(x(2^rP)).
$$
It is a quadratic form, vanishes on torsion points, and satisfies $\widehat h(\phi P)=\deg(\phi)\widehat h(P)$ for every isogeny $\phi$.
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