|
|
|
|
elliptic function
|
(Definition)
|
|
|
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:
Theorem 1 The field of elliptic functions with respect to a lattice $\Lambda$ is generated by $\wp$ and $\wp'$ (the derivative of $\wp$ ).
Proof. See $\cite{lang}$ , chapter 1, theorem 4. 
- 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.
|
"elliptic function" is owned by alozano.
|
|
(view preamble | get metadata)
Cross-references: theorem, derivative, generated by, field, elliptic curve, Liouville's theorem, domain, complex analysis, compact, induce, function, holomorphic, meromorphic, generates, free group, number theory, lattice
There are 18 references to this entry.
This is version 4 of elliptic function, born on 2003-07-22, modified 2003-08-04.
Object id is 4494, canonical name is EllipticFunction.
Accessed 6552 times total.
Classification:
| AMS MSC: | 33E05 (Special functions :: Other special functions :: Elliptic functions and integrals) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|