Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 3 / a / Solution

The cusps are the ${\Gamma }_1(p)$-orbits of primitive columns $(a,c{)}^T$, with simultaneous negation representing the same point of ${\mathbb{P}}^1(\mathbb{Q})$. For the Gamma 1 congruence subgroup, the standard primitive-vector classification separates the orbits according to $d=\gcd (c,p)$. For each divisor $d\mid p$, the two surviving unit coordinates giveorbits when $p>2$. Hence an odd prime gives

For $p=2$, simultaneous negation is already trivial modulo $2$, so the division by two does not apply. In that case ${\Gamma }_1(2)={\Gamma }_0(2)$ has two cusps, represented by infinity and zero. Thus the Number of cusps of Gamma 1 of prime level is