height function
Definition 1
Let $A$ be an abelian group^{}. A height function on $A$ is a function $h\mathrm{:}A\mathrm{\to}\mathrm{R}$ with the properties:

1.
For all $Q\in A$ there exists a constant ${C}_{1}$, depending on $A$ and $Q$, such that for all $P\in A$:
$$h(P+Q)\le 2h(P)+{C}_{1}$$ 
2.
There exists an integer $m\ge 2$ and a constant ${C}_{2}$, depending on $A$, such that for all $P\in A$:
$$h(mP)\ge {m}^{2}h(P){C}_{2}$$ 
3.
For all ${C}_{3}\in \mathbb{R}$, the following set is finite:
$$\{P\in A:h(P)\le {C}_{3}\}$$
Examples:

1.
For $t=p/q\in \mathbb{Q}$, a fraction in lower terms, define $H(t)=\mathrm{max}\{\mid p\mid ,\mid q\mid \}$. Even though this is not a height function as defined above, this is the prototype of what a height function should look like.

2.
Let $E$ be an elliptic curve^{} over $\mathbb{Q}$. The function on $E(\mathbb{Q})$, the points in $E$ with coordinates in $\mathbb{Q}$, ${h}_{x}:E(\mathbb{Q})\to \mathbb{R}$ :
$${h}_{x}(P)=\left\{\genfrac{}{}{0pt}{}{\mathrm{log}H(x(P)),ifP\ne 0}{0,ifP=0}\right\}$$ is a height function ($H$ is defined as above). Notice that this depends on the chosen Weierstrass model of the curve.

3.
The canonical height of $E/\mathbb{Q}$ (due to Neron and Tate) is defined by:
$${h}_{C}(P)=1/2\underset{N\to \mathrm{\infty}}{lim}{4}^{(N)}{h}_{x}([{2}^{N}]P)$$ where ${h}_{x}$ 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 $h\mathrm{:}A\mathrm{\to}\mathrm{R}$ be a height function. Suppose that for the integer $m$, as in property (2) of height, the quotient group^{} $A\mathrm{/}m\mathit{}A$ is finite. Then $A$ is finitely generated^{}.
References
 1 Joseph H. Silverman, The Arithmetic of Elliptic Curves. SpringerVerlag, New York, 1986.
Title  height function 
Canonical name  HeightFunction 
Date of creation  20130322 13:49:09 
Last modified on  20130322 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 