I am trying to solve the following question:
Show that there a meromorphic $1$-form $\omega$ on a torus with
- a double order pole at one point
- a double order zero at another point
- holomorphic elsewhere
- has only real periods.
Hint: Remove a parallelogram from the Euclidean plane and identify opposite edges by translation. Analyze the corresponding $1$-form and deduce how the parallelogram has to look like.
This hint does not make any sense to me. It is too vague for me... I cannot even imagine the space we obtained by the above procedure in hints...
Could anyone give me a more detailed hint or answer? Thanks a lot!
I'm not sure with the condition that $\omega$ has only real periods.
Despite that, there is a construction.
Let $\Lambda = \omega_1 \mathbb{Z} + \omega_2\mathbb{Z}$, $X = \mathbb{C}/\Lambda$. And write the associated Weierstrass elliptic function $\wp(z)$. $$\wp(z) = \frac{1}{z^2}+\sum_{\lambda\in\Lambda-\{0\}}\frac{1}{(z+\lambda)^2}-\frac{1}{\lambda^2}$$ Then $\omega = (\wp(z) - \wp(\omega_1/2))dz$ is meromorphic with a double order pole at $0$, and double order zero at $\omega_1/2$.