elliptic functionEllipticFunction2003-07-22 14:02:392003-08-04 14:54:34Definitionelliptic functionweierstrass% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thm}{Theorem}
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\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}Let $\Lambda \in \mathbb{C}$ be a lattice in the sense of number
theory, i.e. a 2-dimensional free group over ${\mathbb{Z}}$ which
generates $\mathbb{C}$ over $\mathbb{R}$.\\
An {\it elliptic function} $\phi$, with respect to the lattice
$\Lambda$, is a meromorphic funtion $\phi:\mathbb{C} \to
\mathbb{C}$ which is $\Lambda$-periodic:
$$ \phi(z+\lambda)=\phi(z),\quad \forall z\in \mathbb{C},\quad \forall \lambda \in \Lambda$$
{\bf Remark}: An elliptic function which is holomorphic is
constant. Indeed such a function would induce a holomorphic
function on ${\mathbb{C}/\Lambda}$, which is compact (and it is a
standard result from Complex Analysis that any holomorphic
function with compact domain is constant, this follows from
Liouville's Theorem).
{\bf Example}: The Weierstrass $\wp$-function (see elliptic curve)
is an elliptic function, probably the most important. In fact:
\begin{thm} The field of elliptic functions with respect to a lattice
$\Lambda$ is generated by $\wp$ and $\wp'$ (the derivative of
$\wp$).
\end{thm}
\begin{proof}
See $\cite{lang}$, chapter 1, theorem 4.
\end{proof}
\begin{thebibliography}{9}
\bibitem{milne} James Milne, {\em Modular Functions and Modular Forms}, online course notes. \PMlinkexternal{http://www.jmilne.org/math/CourseNotes/math678.html}{http://www.jmilne.org/math/CourseNotes/math678.html}
\bibitem{lang} Serge Lang, {\em Elliptic Functions}. Springer-Verlag, New York.
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.
\end{thebibliography}