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 $C_3\in ℝ$, 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 $H(t)=\max \{\mid p\mid ,\mid q\mid \}$. 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 $\mathbb{Q}$, $h_x:E(\mathbb{Q})\to ℝ$ :

   $$h_x(P)=\left\{\genfrac{}{}{0pt}{}{\log H(x(P)),ifP\ne 0}{0,ifP=0}\right\}$$

   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.

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