canonical height on an elliptic curveCanonicalHeightOnAnEllipticCurve2006-11-08 17:58:092006-11-08 20:41:50Definitionelliptic curvecanonical height% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}Let $E/\Rats$ be an elliptic curve. It is often useful to have a notion of {\it height} of a point, in order to talk about the arithmetic complexity of a point $P$ in $E(\Rats)$. For this, one defines height functions. For example, in $\Rats$ one can define a height by
$$H(p/q)=max(|p|,|q|).$$
Following the example of $\Rats$, one may define a height on $E/\Rats$ 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 $f:E(\Rats)\to \Reals$ on $E(\Rats)$ (i.e. $f(P)=f(-P)$ for any $P\in E(\Rats)$) 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).
\begin{defn}
Let $\Rats$ be a number field and let $E$ be an elliptic curve defined over $\Rats$. The canonical height (or N\'eron-Tate height) on $E/\Rats$, denoted by $\hat{h}$, is the function on $E(\Rats)$ (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 $f:E(\Rats)\to \Reals$.
\end{defn}
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:
\begin{thm}
Let $E/\Rats$ and let $\hat{h}$ be the canonical height on $E$. Then:
\begin{enumerate}
\item 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 $P,Q \in E(\overline{\Rats})$.
\item For all $m\in \Ints$ and all $P\in E(\overline{\Rats})$:
$$\hat{h}([m]P)=m^2\hat{h}(P).$$
\item The height $\hat{h}$ is even and the pairing:
$$\langle \cdot, \cdot \rangle : E(\overline{\Rats})\times E(\overline{\Rats}) \to \Reals,\quad \langle P,Q \rangle = \hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)$$
is bilinear (usually called the N\'eron-Tate pairing on $E/\Rats$).
\item For all $P\in E(\overline{\Rats})$ one has $\hat{h}(P)\geq 0$ and $\hat{h}(P)=0$ if and only if $P$ is a torsion point.
\end{enumerate}
\end{thm}