PlanetMath: canonical height on an elliptic curve

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canonical height on an elliptic curve

(Definition)

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 in $E(\mathbb{Q})$ . For this, one defines height functions. For example, in $\mathbb{Q}$ one can define a height by

$\displaystyle H(p/q)=max(\vert p\vert,\vert q\vert).$

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

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

In fact, given any even function $f:E(\mathbb{Q})\to \mathbb{R}$ on $E(\mathbb{Q})$ (i.e.

for any $P\in E(\mathbb{Q})$ ) one can define a height by:

$\displaystyle h_f(P)=\log H(f(P)).$

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

Definition 1 Let $\mathbb{Q}$ be a number field and let 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:

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

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 is due to J. Tate. In particular, one can simply choose to be the -function, whose degree is . The canonical height satisfies the following properties:

Theorem 1 Let $E/\mathbb{Q}$ and let $\hat{h}$ be the canonical height on . Then:

The height $\hat{h}$ satisfies the parallelogram law:
$\displaystyle \hat{h}(P+Q)+\hat{h}(P-Q)=2\hat{h}(P)+2\hat{h}(Q)$
for all $P,Q \in E(\overline{\mathbb{Q}})$ .
For all $m\in \mathbb{Z}$ and all $P\in E(\overline{\mathbb{Q}})$ :
$\displaystyle \hat{h}([m]P)=m^2\hat{h}(P).$
The height $\hat{h}$ is even and the pairing:
$\displaystyle \langle \cdot, \cdot \rangle : E(\overline{\mathbb{Q}})\times E(\... ...) \to \mathbb{R},\quad \langle P,Q \rangle = \hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)$
is bilinear (usually called the Néron-Tate pairing on $E/\mathbb{Q}$ ).
For all $P\in E(\overline{\mathbb{Q}})$ one has $\hat{h}(P)\geq 0$ and $\hat{h}(P)=0$ if and only if is a torsion point.

"canonical height on an elliptic curve" is owned by alozano.

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Other names:

Neron-Tate height

Also defines:

canonical height

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Cross-references: torsion, bilinear, pairing, even, parallelogram law, degree, real, function, number field, properties, even function, height functions, order, point, height, elliptic curve
There are 2 references to this entry.

This is version 3 of canonical height on an elliptic curve, born on 2006-11-08, modified 2006-11-08.
Object id is 8534, canonical name is CanonicalHeightOnAnEllipticCurve.
Accessed 1501 times total.

Classification:

AMS MSC:	14H52 (Algebraic geometry :: Curves :: Elliptic curves)
	11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields)
	11G07 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over local fields)

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