Showing the cusps in $\Gamma_1(4)$: [$\infty$], [0], [1/2] equal $\mathbb{P_1(Q)}$ I am trying to show that $\Gamma_1(4)$ only has three cusps: $0, 1/2$ and $\infty$. To do so, I need to show $\Gamma_{0},\Gamma_{\infty}, \Gamma_{1/2}$ equal $\mathbb{P_1(Q)}$.
Here is what I have so far:
$\Gamma_{\infty}= \{\frac{4a+1}{4c} |  a,c \in \mathbb{Z}, (4a+1,4c)=1 \}$
$\Gamma_{0}= \{\frac{b}{4d+1} |  b,d \in \mathbb{Z}, (4d+1,b)=1 \}$
$\Gamma_{1/2}= \{\frac{4a+1+2b}{4c+8d+2} |  a,b,c,d \in \mathbb{Z}, (4a+1)(4d+1)-4bc=1 \}=  \{\frac{4a+1}{4c} |  a,b,c,d \in \mathbb{Z}, (4a+1)(4d+1)-4bc=1 \}$
Now how do I show their union equals $\mathbb{P_1(Q)}$?

Let $\frac{r}{s}$ be an element of $\mathbb{Q}$ such that $\gcd(r,s) = 1$.
We shall distinguish the following cases:

*

*$r$ odd, $4\mid s$

*$r$ any number, $s$ odd

*$r$ odd, $s\equiv 2 \mod 4$
Note that if $A \equiv 3 \mod 4$, then $A^2 \equiv 1\mod 4$.
For the first case: If $r\equiv 1 \mod 4$, choose $a = r, b = $anything, $ c = s, d = 1$. Then
$$\begin{pmatrix} a&b\\c&d \end{pmatrix}.\infty = \frac{a}{c} = \frac{r}{s}.$$
Similarly, if $r\equiv 3\mod 4$, then $a = r^2, b = $anything, $c = rs, d = 1$ does the job:
$$\begin{pmatrix} a&b\\c&d \end{pmatrix}.\infty = \frac{a}{c} = \frac{r^2}{sr} = \frac{r}{s}.$$
Now consider the second case: Let $r$ be any integer, $s\equiv 1\mod 4$. Then
$$\begin{pmatrix} a&b\\c&d \end{pmatrix}.0 = \frac{b}{d} = \frac{r}{s}$$
if $a=1,b=r, c=0,d=s$. If $s\equiv 3\mod 4$, choose $a=1,b=rs,c=0,d=s^2$ instead and proceed as above:
$$\begin{pmatrix} a&b\\c&d \end{pmatrix}.0 = \frac{b}{c} = \frac{rs}{s^2} = \frac{r}{s}.$$
Finally, if $s\equiv 2\mod 4$, and $r\equiv 1 \mod 4$, then we can write $s = 4l+2, r = 4k+1$. Hence,
$$\begin{pmatrix} a&b\\c&d \end{pmatrix}.\frac{1}{2} = \frac{a/2+b}{c/2+d} = \frac{1/2+2k}{4l/2+1} = \frac{1+4k}{4l+2} = \frac{r}{s}$$
for $a = 1, b=2k, c=4l, d=1$. If $s\equiv 2\mod 4$ and $r\equiv 3\mod 4$, then we can write $s=4l+2, r = 4k+3$. Therefore,
$$\begin{pmatrix} a&b\\c&d \end{pmatrix}.\frac{1}{2} = \frac{a/2+b}{c/2+d} = \frac{1/2+2k+1}{4l/2+1} = \frac{1+4k+2}{4l+2} = \frac{r}{s}$$
for $a=1,b=2k+1,c=4l,d=1$.
We have accounted for all possible cases for $r$ and $s$ and thus we are done showing that the cusps are $[0],[\frac{1}{2}],$ and $[\infty]$.
