The twice punctured complex plane $\mathbb{C}-\{0,1\}$ has as its universal cover the upper half plane via elliptic modular function.

I am looking for the constructions of the covering map from the upper half plane of the domain $\mathbb{C} - \{ 0, i, 2i, 3i, \cdots\, 1, 1+i, 1+2i, 1+3i, \cdots \}$. Any reference is appreciated.

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    $\begingroup$ What do you want to know about this universal cover? $\endgroup$ Aug 12, 2018 at 13:09
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    $\begingroup$ I want an explicit construction of the covering map or some functional relations like it satisfies certain ODEs. I guess that entire map to this domain need to be periodic and this may be proved by looking at covering map. $\endgroup$ Aug 12, 2018 at 13:29
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    $\begingroup$ The universal cover is still the upper half plane but the Deck transformation group is the free group on infinitely many generators (loops around the infinitely many punctures). You seem to want periodicity under some translations which form an abelian group . $\endgroup$ Aug 12, 2018 at 13:35
  • $\begingroup$ I did not get why you expect any "periodicity". If I understood you correctly you delete the lattice points from the first quadrant only! Can you clarify what is your domain, really? $\endgroup$ Aug 12, 2018 at 21:45


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