Congruence subgroup

A congruence subgroup of $S{L}_n(\mathbb{Z})$ contains the kernel of reduction modulo some positive integer.

The principal congruence subgroup of level $N$ isIt has finite index because the target is finite.

The Gamma 0 congruence subgroup is

The subgroupis a congruence subgroup of the modular group.

For an odd prime $p$, the group ${\Gamma }_1(p)$ has $p-1$ cusps. The general primitive-vector count givesThe exceptional group ${\Gamma }_1(2)={\Gamma }_0(2)$ has two cusps, represented by infinity and zero.

For $p\ge 5$, the Gamma 1 congruence subgroup has $(p-1)/2$ modular cusps of cusp width one and $(p-1)/2$ of cusp width $p$. Primitive columns modulo $p$, taken up to sign, are acted on by $(a,c)\mapsto (a+bc,c)$. The classes with $c=0$ have cusp width one, the others cusp width $p$. There are no effective elliptic stabilizers. The genus formula for a modular curve gives genus $(p-5)(p-7)/24$.