= Canonical height of an elliptic curve
{title2=$\widehat h$}
= Canonical height
{synonym}
= Normalized height on an elliptic curve
{synonym}
= Normalized height
{synonym}
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 is
$$
\widehat h(P)=\frac12\lim_{r\to\infty}4^{-r}h_x([2^r]P).
$$
The 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.<Elliptic Curves by James S. Milne (2nd edition)>[Chapter IV, Section 4, Lemmas 4.4–4.6 and the normalization footnote after Lemma 4.6]{ref}
The estimate $h_x(2P)=4h_x(P)+O(1)$ makes successive terms differ by a summable <geometric series>, proving convergence and $\widehat h(P)=h_x(P)/2+O(1)$. The resulting <quadratic form> satisfies $\widehat h([m]P)=m^2\widehat h(P)$ 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> $\phi:E\to E'$, the corresponding normalized heights satisfy $\widehat h_{E'}(\phi P)=\deg(\phi)\widehat h_E(P)$.
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