Find the cusps for the congruence subgroup $\Gamma_0(p)$. How does one go about doing this? I know the definition of a cusp – the orbit for the action of $G$, in this case $\Gamma_0(p)$, on $\mathbb Q\cup\{\infty\}$, but I don't see how to get myself started thought.
The solution is "the classes of $0$ and $\infty$".
Best Answer
first of all, you can try to see who is in the orbit of $\infty$.
If $z$ is in the orbit of $\infty$ then there exist a matrix on $\Gamma_0(p)$ say $\gamma=\left(\begin{matrix} a & b \\ pc & d\end{matrix}\right)$ such that $\gamma \infty =z$. This implies $\frac{a}{pc}=z$, then the numbers in the orbit of infinity are the rational number of the form $\frac{a}{pc}$ where $\gcd(a,pc)=1$ and $a\neq 0$ (because $ad-pbc=1$) .
On the other hand, it is easy to check that the other rational numbers are in the same orbit, and that $0$ is in that orbit.