height function

Definition 1

Let A be an abelian groupMathworldPlanetmath. A height function on A is a function h:AR with the properties:

  1. 1.

    For all QA there exists a constant C1, depending on A and Q, such that for all PA:

  2. 2.

    There exists an integer m2 and a constant C2, depending on A, such that for all PA:

  3. 3.

    For all C3, the following set is finite:



  1. 1.

    For t=p/q, a fraction in lower terms, define H(t)=max{p,q}. Even though this is not a height function as defined above, this is the prototype of what a height function should look like.

  2. 2.

    Let E be an elliptic curveMathworldPlanetmath over . The function on E(), the points in E with coordinates in , hx:E() :


    is a height function (H is defined as above). Notice that this depends on the chosen Weierstrass model of the curve.

  3. 3.

    The canonical height of E/ (due to Neron and Tate) is defined by:


    where hx is defined as in (2).

Finally we mention the fundamental theorem of “descent”, which highlights the importance of the height functions:

Theorem 1 (Descent)

Let A be an abelian group and let h:AR be a height function. Suppose that for the integer m, as in property (2) of height, the quotient groupMathworldPlanetmath A/mA is finite. Then A is finitely generatedMathworldPlanetmathPlanetmath.


Title height function
Canonical name HeightFunction
Date of creation 2013-03-22 13:49:09
Last modified on 2013-03-22 13:49:09
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Definition
Classification msc 14H52
Related topic EllipticCurve
Related topic RankOfAnEllipticCurve
Related topic ArithmeticOfEllipticCurves
Related topic CanonicalHeightOnAnEllipticCurve
Defines height function
Defines canonical height
Defines descent theorem