PlanetMath: height function

(more info)

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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.