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# canonical height on an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve. It is often useful to have a notion of height of a point, in order to talk about the arithmetic complexity of a point $P$ in $E(\mathbb{Q})$. For this, one defines height functions. For example, in $\mathbb{Q}$ one can define a height by

$H(p/q)=max(|p|,|q|).$ |

Following the example of $\mathbb{Q}$, one may define a height on $E/\mathbb{Q}$ by

$h_{x}(P)=\begin{cases}\log H(x(P))&\text{if }P\neq O\\ 0&\text{if }P=O.\end{cases}$ |

In fact, given any even function $f:E(\mathbb{Q})\to\mathbb{R}$ on $E(\mathbb{Q})$ (i.e. $f(P)=f(-P)$ for any $P\in E(\mathbb{Q})$) one can define a height by:

$h_{f}(P)=\log H(f(P)).$ |

However, one can refine this definition so that the height function satisfies some very nice properties (see below).

###### Definition.

Let $\mathbb{Q}$ be a number field and let $E$ be an elliptic curve defined over $\mathbb{Q}$. The canonical height (or Néron-Tate height) on $E/\mathbb{Q}$, denoted by $\hat{h}$, is the function on $E(\mathbb{Q})$ (with real values) defined by:

$\hat{h}(P)=\frac{1}{\deg f}\lim_{{N\to\infty}}\frac{h_{f}([2^{N}]P)}{4^{N}}$ |

for any even function $f:E(\mathbb{Q})\to\mathbb{R}$.

The fact that the definition does not depend on the choice of even function $f$ is due to J. Tate. In particular, one can simply choose $f$ to be the $x$-function, whose degree is $2$. The canonical height satisfies the following properties:

###### Theorem.

Let $E/\mathbb{Q}$ and let $\hat{h}$ be the canonical height on $E$. Then:

1. The height $\hat{h}$ satisfies the parallelogram law:

$\hat{h}(P+Q)+\hat{h}(P-Q)=2\hat{h}(P)+2\hat{h}(Q)$ for all $P,Q\in E(\overline{\mathbb{Q}})$.

2. For all $m\in\mathbb{Z}$ and all $P\in E(\overline{\mathbb{Q}})$:

$\hat{h}([m]P)=m^{2}\hat{h}(P).$ 3. The height $\hat{h}$ is even and the pairing:

$\langle\cdot,\cdot\rangle:E(\overline{\mathbb{Q}})\times E(\overline{\mathbb{Q% }})\to\mathbb{R},\quad\langle P,Q\rangle=\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)$ is bilinear (usually called the Néron-Tate pairing on $E/\mathbb{Q}$).

4. For all $P\in E(\overline{\mathbb{Q}})$ one has $\hat{h}(P)\geq 0$ and $\hat{h}(P)=0$ if and only if $P$ is a torsion point.

## Mathematics Subject Classification

11G07*no label found*11G05

*no label found*14H52

*no label found*

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