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# height function

###### Definition 1.

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

1. For all $Q\in A$ there exists a constant $C_{1}$, depending on $A$ and $Q$, such that for all $P\in A$:

$h(P+Q)\leq 2h(P)+C_{1}$ 2. There exists an integer $m\geq 2$ and a constant $C_{2}$, depending on $A$, such that for all $P\in A$:

$h(mP)\geq m^{2}h(P)-C_{2}$ 3. For all $C_{3}\in\mathbb{R}$, the following set is finite:

$\{P\in A:h(P)\leq C_{3}\}$

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 $\mathbb{Q}$, $h_{x}\colon E(\mathbb{Q})\to\mathbb{R}$ :

$h_{x}(P)={{\log H(x(P)),\quad if\ P\neq 0}\choose{0,\quad if\ P=0}}$ 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:$h_{C}(P)=1/2\lim_{{N\to\infty}}4^{{(-N)}}h_{x}([2^{N}]P)$ 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 $h\colon A\to\mathbb{R}$ be a height function. Suppose that for the integer $m$, as in property (2) of height, the quotient group $A/mA$ is finite. Then $A$ is finitely generated.

# References

- 1 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

## Mathematics Subject Classification

14H52*no label found*

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