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Pairing curves on a Riemann Surface

(I know the question is pretty specific)
Now, the picture say it all and I will try to explain it as best as I can.
Consider the following figure,

Each of the pair of circles (and the red lines) define a Riemann surface $\Sigma$.
The top left model is the unit disc and the bottom right is the Riemann surface of the complex logarithm.
The $A$, $B$ and $C$ match the letters in the picture.

So, my question is, what is the relationship between these two surfaces?


Each is obtained as the quotient of the unit

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