Doubly periodic function

Definition

A meromorphic function $f:\mathbb{C}\to S^2$ is termed doubly periodic or elliptic if there exist two real-linearly independent nonzero complex numbers ${\omega }_1$ and ${\omega }_2$ such that:

$f(z+{\omega }_1)=f(z),f(z+{\omega }_2)=f(z)\mathrm{\forall }z\in \mathbb{C}$

The additive subgroup generated by ${\omega }_1$ and ${\omega }_2$ is a lattice in $\mathbb{C}$, and an elliptic function on $\mathbb{C}$ can be defined as a function that descends to a function on the quotient of $\mathbb{C}$ by the lattice.