Meromorphic Function is Elliptic iff Doubly Periodic

Theorem

Let $f: \C \to \C$ be a meromorphic function.

Then $\map f z$ is an elliptic function if and only if it is also doubly periodic.

Historical Note

Meromorphic Function is Elliptic iff Doubly Periodic was discovered by Niels Henrik Abel‎ during his exploration of elliptic functions.