elliptic function

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 elliptic function $\phi$ , with respect to the lattice $\Lambda$ , is a meromorphic funtion $\phi:\mathbb{C} \to \mathbb{C}$ which is $\Lambda$ -periodic: $$ \phi(z+\lambda)=\phi(z),\quad \forall z\in \mathbb{C},\quad \forall \lambda \in \Lambda$$

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).

Example: The Weierstrass $\wp$ -function (see elliptic curve) is an elliptic function, probably the most important. In fact:

Proof. See $\cite{lang}$ , chapter 1, theorem 4.

Bibliography

1

James Milne, Modular Functions and Modular Forms, online course notes. http://www.jmilne.org/math/CourseNotes/math678.html

2

Serge Lang, Elliptic Functions. Springer-Verlag, New York.

3

Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.