canonical height on an elliptic curve

Let E/ be an elliptic curveMathworldPlanetmath. It is often useful to have a notion of height of a point, in order to talk about the arithmetic complexity of a point P in E(). For this, one defines height functions. For example, in one can define a height by


Following the example of , one may define a height on E/ by

hx(P)={logH(x(P))if PO0if P=O.

In fact, given any even function f:E() on E() (i.e. f(P)=f(-P) for any PE()) one can define a height by:


However, one can refine this definition so that the height function satisfies some very nice properties (see below).


Let Q be a number field and let E be an elliptic curve defined over Q. The canonical height (or Néron-Tate height) on E/Q, denoted by h^, is the functionMathworldPlanetmath on E(Q) (with real values) defined by:


for any even function f:E(Q)R.

The fact that the definition does not depend on the choice of even function f is due to J. Tate. In particular, one can simply choose f to be the x-function, whose degree is 2. The canonical height satisfies the following properties:


Let E/Q and let h^ be the canonical height on E. Then:

  1. 1.

    The height h^ satisfies the parallelogram law:


    for all P,QE(¯).

  2. 2.

    For all m and all PE(¯):

  3. 3.

    The height h^ is even and the pairing:


    is bilinear (usually called the Néron-Tate pairing on E/).

  4. 4.

    For all PE(¯) one has h^(P)0 and h^(P)=0 if and only if P is a torsion point.

Title canonical height on an elliptic curve
Canonical name CanonicalHeightOnAnEllipticCurve
Date of creation 2013-03-22 16:23:20
Last modified on 2013-03-22 16:23:20
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Definition
Classification msc 11G07
Classification msc 11G05
Classification msc 14H52
Synonym Neron-Tate height
Related topic HeightFunction
Related topic RegulatorOfAnEllipticCurve
Defines canonical height