elliptic function

Let $\mathrm{\Lambda }\in ℂ$ be a lattice in the sense of number theory, i.e. a 2-dimensional free group over $\mathbb{Z}$ which generates $ℂ$ over $ℝ$.

An elliptic function $\varphi$, with respect to the lattice $\mathrm{\Lambda }$, is a meromorphic funtion $\varphi :ℂ\to ℂ$ which is $\mathrm{\Lambda }$-periodic:

$$\varphi (z+\lambda )=\varphi (z),\forall z\in ℂ,\forall \lambda \in \mathrm{\Lambda }$$

Remark: An elliptic function which is holomorphic is constant. Indeed such a function would induce a holomorphic function on $ℂ/\mathrm{\Lambda }$, 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 $\mathrm{\wp }$-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 $\mathrm{\Lambda }$ is generated by $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$ (the derivative of $\mathrm{\wp }$).

Proof.

See [2], chapter 1, theorem 4. ∎

Title	elliptic function
Canonical name	EllipticFunction
Date of creation	2013-03-22 13:47:03
Last modified on	2013-03-22 13:47:03
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	7
Author	alozano (2414)
Entry type	Definition
Classification	msc 33E05
Related topic	ExamplesOfEllipticFunctions
Related topic	WeierstrassSigmaFunction
Related topic	ModularDiscriminant
Related topic	WeierstrassWpFunction
Related topic	TableOfMittagLefflerPartialFractionExpansions
Related topic	PeriodicFunctions