elliptic function

Let $\mathrm{\Lambda }\in ℂ$ be a lattice in the sense of numbertheory, i.e. a 2-dimensional free group over $\mathbb{Z}$ whichgenerates $ℂ$ 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 isconstant. Indeed such a function would induce a holomorphicfunction on $ℂ/\mathrm{\Lambda }$, which is compact (and it is astandard result from Complex Analysis that any holomorphicfunction with compact domain is constant, this follows fromLiouville’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.∎