Hi all! In this post, we’ll show how we can use the Riemann-Hurwitz formula to derive the genus of the modular curve . But what is ?
The group acts on the complex upper half-plane, denoted . Let be a congruence subgroup of this group, i.e., a group containing the principal congruence subgroup of level N, , defined as The quotient is a noncompact Riemann surface, denoted . Its compactification, denoted , is the quotient . We remark that is called the extended complex upper half-plane, and is given a topology by choosing a particular basis. The cusps of are elements of the orbit of the action of on .
If , the principal congruence subgroup of level N, then is denoted . If , is isomorphic as a Riemann surface to . Now, , so . It therefore has genus . Now, we have a map coming from the inclusion . We will prove, using the Riemann-Hurwitz formula, the following equation for :
Where is the number of elliptic points of order , and is the number of cusps.
How do we go about proving this? First recall the Riemann-Hurwitz formula: if is a surjective morphism of Riemann surfaces of degree , let denote the ramification degree at . Then
There’s a statement in the following proof which I don’t understand, and would like to. If is Galois, the in the fibers of each is equal, this becomes
The map is Galois, so the indices are the same for over a . The degree of this map is (the number of preimages of generic points of ). If , this is explicitly .
We need to compute the ramification degree . The fiber of over is the -equivalence class of cusps, which, by the orbit stabilizer theorem, number where is the stabilizer of (a coset) in (which is isomorphic to ). Thus is the subgroup consisting of matrices such that , and . Consequently, .
Cool. Now, the ramfications of are over , , and . For the map in consideration, and (I don’t think I understand why, perhaps it’s a simple error on my part). Thus, because is Galois, using the Riemann-Hurwitz formula, we get:
Solving for gives us the required formula. QED.
Here are the first few values of : , , , . It’s some fun stuff, and I hope to continue to learn more of it! If anyone can tell me why and , it would be greatly appreciated!