In this question it was described how to realise a torus as the double of an annulus Explicit construction of mirror surface and complex double for an annulus.
In short, the torus is realised considering a complex plane from which two non intersecting disks $D$ and $\overline{D}$, one the complex conjugate of the other, have been removed. Furthermore, the boundary circles $C$ and $\overline{C}$ are glued together. Let $T$ be the torus obtained in such a way.
I would like to understand how to build a meromorphic differential on a torus realised in such way. I think that a naive differential defined as $$\omega = \phi(w) dw,$$ where $w$ is the coordinate on $\mathbb{C}$, fails to be a well defined differential on the torus $T$ unless $\phi$ satisfy some gluing condition, perhaps something like $\phi(\overline{w}) = \phi(w)$ for $w \in C$?
The torus $T$ can be realised also as $\mathbb{C}/{\mathbb{Z} + \tau \mathbb{Z}}$ for a suitable $\tau = \tau(C)$. If $z$ is a coordinate on $T$ realised in this way, $dz$ is an holomorphic differential. Furthermore, if we explicitly compute $\tau(C)$ we can find the change of coordinates $z(w)$ and therefore we can represent the holomorphic differential $dz$ in $w$ coordinates. I did the computation for a particularly simple case, where $C$ is a circle centered on the imaginary axis such that $i \in D$, and I found the following expression $$dz = \frac{dw}{\pi(w^2+1)}.$$
Which does not satisfy the naive gluing condition I would expect, i.e.$\phi(\overline{w}) = \phi(w)$ for $w \in C$.
My questions are
1) What is the correct gluing condition?
2) Is my expression for the holomorphic differential correct?
3) How can I build meromorphic differentials with simple poles? In particular, I would like to know if it is possible to do so withouth using Jacobi theta functions.
