In the above pdf, we calculate the genus of some Riemann surfaces, quotients of the upper half plane by , where is some prime.
The idea is to project the Riemann surface to and to apply the Riemann Hurwitz formula to this map. I include a picture that allows one to visualize the this projection map, which is basically just an appropriate folding of parts of the complex plane.
This post arose after making these calculations one afternoon with Ravi Donepudi, a fellow number theory student here at UIUC. Also, I would like to thank Junxian Li (also in UIUC number theory) for providing the picture and useful discussions.