height functionHeightFunction2003-08-04 13:47:582003-08-05 09:45:26Definitionheight functioncanonical heightdescent theoremheightdescent% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}\begin{defn}
Let $A$ be an abelian group. A height function on $A$ is a
function $h\colon A\to \mathbb{R}$ with the properties:
\begin{enumerate}
\item For all $Q\in A$ there exists a constant $C_1$, depending on
$A$ and $Q$, such that for all $P\in A$:
$$ h(P+Q)\leq 2h(P) +C_1$$
\item There exists an integer $m \geq 2$ and a constant $C_2$,
depending on $A$, such that for all $P\in A$:
$$ h(mP)\geq m^2h(P) -C_2$$
\item For all $C_3 \in \mathbb{R}$, the following set is finite:
$$\{ P\in A: h(P)\leq C_3\} $$
\end{enumerate}
\end{defn}
{\bf Examples}:
\begin{enumerate}
\item 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. \item Let $E$ be an
elliptic curve over $\mathbb{Q}$. The function on $E(\mathbb{Q})$,
the points in $E$ with coordinates in $\mathbb{Q}$,
$h_x\colon E(\mathbb{Q})\to \mathbb{R}$ :
$$h_x(P)={{\log H(x(P)),\quad if\ P\neq 0} \brace {0,\quad if\
P=0}}$$ is a height function ($H$ is defined as above). Notice that
this depends on the chosen Weierstrass model of the curve. \item
The \emph{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).
\end{enumerate}
Finally we mention the fundamental theorem of ``descent'', which
highlights the importance of the height functions:
\begin{thm}[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.
\end{thm}
\begin{thebibliography}{9}
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.
\end{thebibliography}