Let $f:\mathbb{C}\to \mathbb{C}$ be meromorphic, and let $P$ be the set of periods of $f$. Then $P$ satisfies the following:

1. The set of periods has no accumulation points.
2. $(P,+)$ is an abelian group
3. The "interval" $[-\omega ,\omega ]$ contains finitely many periods, so we can choose a smallest one, say ${\omega }_1$. Using a division algorithm, we can show that every period on the line spanned by $\mathbb{R}\omega$ is a multiple of ${\omega }_1$.
4. If $w^{\prime }\in P$ is another period not on the line $\mathbb{R}\omega$, then we can find ${\omega }_2\in P$ such that, all of the periods are of the form $n{\omega }_1+m{\omega }_2$
5. Suppose $W_1=a{\omega }_1+b{\omega }_2$ and $W_2=c{\omega }_1+d{\omega }_2$. Then $W_1,W_2$ span $P$ iff $det(ad-bc)=\pm 1$. That is, the area of the parallelogram $0,{\omega }_1,{\omega }_2,{\omega }_1+{\omega }_2$ has the same unsigned area as $0,W_1,W_2,W_1+W_2$.