Congruence subgroup

Summary

A congruence subgroup of the modular group $\mathrm{SL}_2(\Z)$ is a subgroup defined by congruence conditions on the matrix entries modulo some integer $N$. The main examples are $\Gamma(N)$, $\Gamma_1(N)$ and $\Gamma_0(N)$. Modular forms of level $N$ are modular forms for these groups. Most modular forms that arise in arithmetic, such as those attached to elliptic curves, have a congruence subgroup as their symmetry group.