I'm currently reading some of the geometric theory behind the theory of modular forms, Diamond and Shurman's book is my main reference. If $\Gamma$ is a congruence subgroup of $SL_2(\mathbb{Z})$, the modular curve $Y(\Gamma)$ is defined to be the quotient $\Gamma \backslash \mathfrak{H}$. In order to compactify this curve one needs to add a finite number of cusps $s_1,\dots, s_n$ which are the distinct orbits of $\Gamma$ acting on $\mathbb{Q}\cup \{ \infty\}$.
The case where $\Gamma=SL_2(\mathbb{Z})$ makes sense to me since the group acts transivity on $\mathbb{Q}\cup \infty$, but for general $\Gamma$, why do we need to add in the other orbits (apart from the orbit containing $\infty$) in order to compactify the curve?
A somewhat related question: Diamond and Shurman describe a modular form of weight $k$ for $\Gamma$ as being holomorphic at the "limit points" and yet I cannot find any other reference to this idea. How does one realize these cusps as "limit points"?
Any help would be appreciated!
For the first question, you might find this elementary example helpful:
For $ p $ a prime, the modular curve $ X_{0}(p) = \mathcal{H} / \Gamma_{0} $ has two cusps on $ \mathbb{P}(\mathbb{Q}) $ which can be denoted by $ [1,0] $ and $ [0,1] $. To see this, we partition rational numbers $ \mathbb{Q} $ as $ \{ t \in \mathbb{Q} : p|deno(t) \} \cup (\{ t \in \mathbb{Q}: p \not | deno(t) \} \backslash \{ 0 \}) $. Here by $ deno(t) $ I meant denominator of t.
For the first set, let $ r = u/v $ where $ p | v $. Because of the irreducibility of the representation of $ r $, there must be $ c, d \in \mathbb{Z} $ such that $ cu - dv = 1 $. The matrix $ M=(-c,d;v,-u) \in \Gamma_{0}$ takes $ r $ to $ \infty $, i.e., $ \frac {-cr + d}{vr - u} = \infty $. Similarly for a given $ s = u/v $ you can always find a matrix in $ \Gamma_{0} $ taking $ \infty $ to $ s $.
For the second set, let $ r = u/v $ where $ p \not |v $. From $ gcd(u,v)=1$ and $ gcd(p,v)=1 $ we have that $ gcd(v, pu)=1 $ therefore, we have another determinant-form expression $ cpu - dv = 1 $ for some $ c,d \in \mathbb{Z} $. The matrix $ M=(v,-u;cp,-d) \in \Gamma_{0} $ is the matrix taking $ r $ to zero and similarly for a given $ s=u/v $ you can find a matrix in $ \Gamma_{0} $ taking a given $ 0 $ to $ s $. it simply shows that the modular curve $ X_{0}(p) $ for a prime number $ p $ has two cusps which can be roughly represented by $ 0 $ and $ \infty $.