rank of an elliptic curveRankOfAnEllipticCurve2003-08-04 14:32:522007-05-26 15:51:18Definitionweak Mordell-Weil theoremrank of an elliptic curvemordellweilrankheight% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}Let $K$ be a number field and let $E$ be an elliptic curve over
$K$. By $E(K)$ we denote the set of points in $E$ with coordinates
in $K$.
\begin{thm}[Mordell-Weil]$E(K)$ is a finitely generated abelian
group.
\end{thm}
\begin{proof}
The proof of this theorem is fairly involved. The
main two ingredients are the so called ``weak Mordell-Weil theorem''
(see below), the concept of height function for abelian groups and
the ``descent'' theorem. \\See $\cite{silverman}$, Chapter VIII, page
189.
\end{proof}
\begin{thm}[Weak Mordell-Weil]$E(K)/mE(K)$ is
finite for all $m\geq 2$.
\end{thm}
The Mordell-Weil theorem implies that for any elliptic curve $E/K$
the group of points has the following structure:
$$E(K)\simeq E_{\operatorname{torsion}}(K)\bigoplus {\mathbb{Z}}^R$$
where $E_{\operatorname{torsion}}(K)$ denotes the set of points of finite order (or torsion group),
and $R$ is a non-negative integer which is called the $rank$ of the
elliptic curve. It is not known how big this number $R$ can get
for elliptic curves over $\mathbb{Q}$. The largest rank known for
an elliptic curve over $\mathbb{Q}$ is 28 \PMlinkexternal{Elkies (2006)}{http://www.math.hr/~duje/tors/tors.html}.
Note: see Mazur's theorem for an account of the possible torsion subgroups over $\mathbb{Q}$.
{\bf Examples}:
\begin{enumerate}
\item The elliptic curve $E_1/\mathbb{Q}\colon y^2=x^3+6$ has rank 0
and $E_1(\mathbb{Q})\simeq {0}$.
\item Let $E_2/\mathbb{Q}\colon y^2=x^3+1$, then
$E_2(\mathbb{Q})\simeq \mathbb{Z}/6\mathbb{Z}$. The torsion group
is generated by the point $(2,3)$.
\item Let $E_3/\mathbb{Q}\colon y^2=x^3+109858299531561$, then
$E_3(\mathbb{Q})\simeq \mathbb{Z}/3\mathbb{Z}\bigoplus
{\mathbb{Z}}^5$. See
\PMlinkexternal{generators}{http://math.bu.edu/people/alozano/Torsion.html}
here.
\item Let $E_4/\mathbb{Q}\colon y^2 +1951/164xy
-3222367/40344y=x^3+3537/164x^2-40302641/121032x$, then
$E_4(\mathbb{Q})\simeq {\mathbb{Z}}^{10}$. See
\PMlinkexternal{generators}{http://math.bu.edu/people/alozano/Examples.html}
here.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{milne} James Milne, {\em Elliptic Curves}, online course notes. \PMlinkexternal{http://www.jmilne.org/math/CourseNotes/math679.html}{http://www.jmilne.org/math/CourseNotes/math679.html}
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.
\bibitem{silverman2} Joseph H. Silverman, {\em Advanced Topics in
the Arithmetic of Elliptic Curves}. Springer-Verlag, New York,
1994.
\bibitem{shimura} Goro Shimura, {\em Introduction to the
Arithmetic Theory of Automorphic Functions}. Princeton University
Press, Princeton, New Jersey, 1971.
\end{thebibliography}