canonical height on an elliptic curve
Let be an elliptic curve. It is often useful to have a notion of height of a point, in order to talk about the arithmetic complexity of a point in . 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 by
In fact, given any even function on (i.e. for any ) one can define a height by:
However, one can refine this definition so that the height function satisfies some very nice properties (see below).
Definition.
Let be a number field and let be an elliptic curve defined over . The canonical height (or Néron-Tate height) on , denoted by , is the function on (with real values) defined by:
for any even function .
The fact that the definition does not depend on the choice of even function is due to J. Tate. In particular, one can simply choose to be the -function, whose degree is . The canonical height satisfies the following properties:
Theorem.
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 |