height function

Definition 1

Let $A$ be an abelian group. A height function on $A$ is afunction $h\mathrm{:}A\mathrm{\to }\mathrm{R}$ 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 thisis not a height function as defined above, this is the prototypeof what a height function should look like.

2. Let $E$ be anelliptic 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 thatthis 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”, whichhighlights 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 $A\mathrm{/}m\mathit{}A$ isfinite. Then $A$ is finitely generated.