PlanetMath: height function

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

height function

(Definition)

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

For all $Q\in A$ there exists a constant , depending on and , such that for all $P\in A$ :
$\displaystyle h(P+Q)\leq 2h(P) +C_1$
There exists an integer $m \geq 2$ and a constant , depending on , such that for all $P\in A$ :
$\displaystyle h(mP)\geq m^2h(P) -C_2$
For all $C_3 \in \mathbb{R}$ , the following set is finite:
$\displaystyle \{ P\in A: h(P)\leq C_3\}$

Examples:

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.
Let be an elliptic curve over $\mathbb{Q}$ . The function on $E(\mathbb{Q})$ , the points in with coordinates in $\mathbb{Q}$ , $h_x\colon E(\mathbb{Q})\to \mathbb{R}$ :
$\displaystyle h_x(P)={{\log H(x(P)),\quad if\ P\neq 0} \brace {0,\quad if P=0}}$
is a height function ( is defined as above). Notice that this depends on the chosen Weierstrass model of the curve.
The canonical height of $E/\mathbb{Q}$ (due to Neron and Tate) is defined by:
$\displaystyle h_C(P)=1/2 \lim_{N\to \infty} 4^{(-N)}h_x([2^N]P)$
where 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 be an abelian group and let $h\colon A \to \mathbb{R}$ be a height function. Suppose that for the integer , as in property (2) of height, the quotient group is finite. Then is finitely generated.