Mordell-Weil theoremMordellWeilTheorem2002-02-03 01:34:422005-03-01 13:09:29Theorem% 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 \PMlinkname{weak Mordell-Weil theorem}{WeakMordellWeilTheorem}, the concept of height function for abelian groups and
the ``\PMlinkname{descent}{HeightFunction}'' theorem. \\See $\cite{silverman}$, Chapter VIII, page
189.
\end{proof}
\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}