How to count to cusps of the modular curve $X_1(N)$, i.e., for the congruence subgroup $\Gamma_1(N)$

Question

I already saw the calculations in the book of Diamond and Shurman pag 103 to count the number of cusps of $\Gamma_0(N)$ $(N>1)$ but I really can not understand how to do the calculations for $\Gamma_1(N)$ with $(N>1)$.

The calculation of the book is:

The To count the cusps of $\gamma_0(N)$ recall from Proposition 3.8.3 that for this group, vectors $\begin{bmatrix} a\\ c \end{bmatrix}$ and $\begin{bmatrix} a'\\ c' \end{bmatrix}$ with $\gcd(a,c)=\gcd(a′,c′)=1$ represent the same cusp when $\begin{bmatrix} ya'\\ c' \end{bmatrix}\equiv \begin{bmatrix} a+jc\\ c \end{bmatrix} \mod N$ for some $j$ and $y$ with $\gcd(y,N)=1$. The bottom condition, $c′\equiv yc \mod N$ for some such $y$, is equivalent to $\gcd(c′,N)=\gcd(c,N)$, in wich case letting $d=\gcd(c,N)$ and letting $y_0\in\mathbb{Z}$ satisfy $y_0\equiv c′c^{−1} \mod N$ makes the condition equivalent to $y\equiv y_0+iN/d \mod N$ for some $i$ (confirming the calculations in the paragraph is Exercise 3.8.4.). For any divisor $d$ of $N$, pick one value $c$ modulo $N$ such that $\gcd(c,n)=d$. Then any cusp of $\Gamma_0(N)$ represented by some vector $\begin{bmatrix} a\\ c \end{bmatrix}$ with $\gcd(c′,N)=d$ is also represented by $\begin{bmatrix} a\\ c \end{bmatrix}$ whenever $(y_0+iN/d)a′\equiv a+jc\mod N$ for some $i$ and $j$ , or $a\equiv y_0a′\mod \gcd(c,N,a′N/d)$, or $a\equiv y_0a′\mod \gcd(d,N/d)$. Also, $a$ is relatively prime to $d$ since $\gcd(a,d)∣\gcd(a,c)=1$, so $a$ is relatively prime to $\gcd(d,N/d)$. Thus for each divisor $d$ of $N$ there are $\phi(\gcd(d,N/d))$ cusps, and the number of cusps of $\Gamma_0(N)$ is therefore $\sum _{d∣N}\phi(\gcd(d,N/d))$.

There is an alternative calculation which is easier and very understandable for the cusps of $\Gamma_0(N)$ in How does this argument to count the cusps of $Γ_0 (N)$ work? but I do not understand how to used to get the number of the cusps of $\Gamma_1(N), N>1$.

Any help will be very very important, thank you in advance.

Answer 1

A good reference is Miyake, Modular forms (Springer, 1989), Section 4.2. There is an explicit bijection between the set of cusps of $X_1(N)(\mathbf{C})$ and the set \begin{equation*} \bigl\{(c,d) : c \in \mathbf{Z}/N\mathbf{Z}, \, d \in (\mathbf{Z}/(c,N)\mathbf{Z})^\times \bigr\}/\{\pm 1\}. \end{equation*} Explicitly, for $\gamma = (\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}) \in \mathrm{SL}_2(\mathbf{Z})$, the cusp $\Gamma_1(N) \cdot \frac{a}{c} = \Gamma_1(N) \cdot \gamma \infty \in \Gamma_1(N) \backslash \mathbf{P}^1(\mathbf{Q})$ corresponds to the class of the pair $(c,d)$. From this one can compute the number of cusps of $X_1(N)(\mathbf{C})$. For $N=1,2,4$, the modular curve $X_1(N)(\mathbf{C})$ has $1,2,3$ cusps respectively, and for $N \neq 1,2,4$, the number of cusps of $X_1(N)(\mathbf{C})$ is \begin{equation*} \frac{N}{2} \prod_{p | N} \Bigl( 1- p^{-2} + v_p(N)(1-p^{-1})^2 \Bigr) \end{equation*} where the product is over the prime divisors $p$ of $N$, and $v_p(N)$ denotes the $p$-adic valuation of $N$. (This can be deduced from Theorem 4.2.9 in Miyake, using the fact that the RHS of that theorem involves a multiplicative function of $N$.)

For example, if $N=p$ is prime, the modular curve $X_1(p)(\mathbf{C})$ has $(p-1)$ cusps: half of them are above the cusp $\infty \in X_0(p)(\mathbf{Q})$, and the other half above the cusp $0 \in X_0(p)(\mathbf{Q})$.

The set of cusps of $X_1(N)(\mathbf{C})$ is endowed with an action of the Galois group $\mathrm{Aut}(\mathbf{C}/\mathbf{Q})$ and it is easier to cound the number of Galois orbits. Namely, a complete set of representatives of the Galois orbits of cusps in $X_1(N)(\mathbf{C})$ is given by the cusps $\frac{1}{k}$ with $0 \leq k \leq \lfloor \frac{N}{2} \rfloor$.