Let be a lattice in the sense of number theory, i.e. a 2-dimensional free group over which generates over .

An elliptic function , with respect to the lattice , is a meromorphic funtion which is -periodic:

Remark: An elliptic function which is holomorphic is constant. Indeed such a function would induce a holomorphic function on , 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 -function (see elliptic curve) is an elliptic function, probably the most important. In fact:

Theorem 1   The field of elliptic functions with respect to a lattice is generated by and (the derivative of ).

Proof. See , 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.