Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.

If $f:\mathbb{C} \to \mathbb{C}$ is elliptic, then there exists nonequal $t_0,t_1 \in \mathbb{C}$ such that
$$ f(z) = f(z + t_0), f(z) = f(z + t_1)
$$
the parallelogram $0,t_0,t_0+t_1,t_1$ is called the **fundamental parallelogram** of $f$. $f$'s value is entirely determined by its values on the fundamental parallelogram. By Liouville's theorem, any holomorphic elliptic function must be constant, so the usual elliptic functions are meromorphic. In fact, they must have at least two poles (with multiplicity) on the fundamental parallelogram.