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

In fact, given any even function $f:E(\mathbb{Q})\to ℝ$ 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 $\mathrm{Q}$ be a number field and let $E$ be an elliptic curve defined over $\mathrm{Q}$. The canonical height (or Néron-Tate height) on $E\mathrm{/}\mathrm{Q}$, denoted by $\widehat{h}$, is the function on $E\mathit{}\mathrm{(}\mathrm{Q}\mathrm{)}$ (with real values) defined by:

$$\widehat{h}(P)=\frac{1}{\mathrm{deg}f}\underset{N\to \mathrm{\infty }}{lim}\frac{h_f([2^N]P)}{4^N}$$

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

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 $E\mathrm{/}\mathrm{Q}$ and let $\widehat{h}$ be the canonical height on $E$. Then:

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

   $$\widehat{h}(P+Q)+\widehat{h}(P-Q)=2\widehat{h}(P)+2\widehat{h}(Q)$$

   for all $P,Q\in E(\overline{\mathbb{Q}})$.

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

   $$\widehat{h}([m]P)=m^2\widehat{h}(P).$$

3. The height $\widehat{h}$ is even and the pairing:

   $$⟨\cdot ,\cdot ⟩:E(\overline{\mathbb{Q}})\times E(\overline{\mathbb{Q}})\to ℝ,⟨P,Q⟩=\widehat{h}(P+Q)-\widehat{h}(P)-\widehat{h}(Q)$$

   is bilinear (usually called the Néron-Tate pairing on $E/\mathbb{Q}$).

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