Give an explicit description of the covering space of the torus corresponding to the subgroup of generated by .
You know that the universal cover of is and consists of translations by elements of the lattice . Your subgroup corresponds to the subgroup consisting of . Your covering space then corresponds to the quotient , where (as I hope you know) this means that if they differ by a -action.
Right! Intuitively what we've done is decomposed ( and ) which induces a decomposition (where are the deck transformation and respectively). Intuitively you can think then of as where the identifcation (our identification) corresponds to taking and wrapping it around itself (horizontally) to get a cylinder, and then modding by gives you the vertical identifications giving us the torus.
So, in short, yes. It's the cylinder.