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Canonical Height

Let E be an elliptic curve over a number field K, and let h be the absolute logarithmic height. For a nonidentity point P in E(K), put h_E(P)=1/2h(x(P)), with h_E(O)=0. The canonical height, also called the Néron-Tate height, is

h^^(P)=lim_(n->infty)(h_E([n]P))/(n^2).

It differs from h_E by a bounded function but satisfies the exact quadratic relation h^^([n]P)=n^2h^^(P). Over a number field, h^^(P)=0 if and only if P is a torsion point.

The associated Néron-Tate height pairing is

<P,Q>=1/2(h^^(P+Q)-h^^(P)-h^^(Q)).

It induces a positive definite inner product on the group of K-rational points modulo torsion. The Gross-Zagier formula expresses a derivative of a Rankin L-series in terms of such a height.

See also

Elliptic Curve, Gross-Zagier Formula, Group Torsion, Number Field, Rankin L-Series, Torsion Point

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References

Néron, A. "Quasi-fonctions et hauteurs sur les variétés abéliennes." Ann. Math. 82, 249-331, 1965.Silverman, J. H. The Arithmetic of Elliptic Curves, 2nd ed. New York: Springer, 2009. https://doi.org/10.1007/978-0-387-09494-6.

Cite this as:

Weisstein, Eric W. "Canonical Height." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CanonicalHeight.html

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