---
tag: 1H6R856P3C
kind: concept
name: Congruence subgroup
msc: [11F06, 20H05]
specializes: [0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence_subgroup
same_as:
  wikidata: Q5161000
  wikipedia: Congruence subgroup
---

## Summary

A congruence subgroup of the [[modular group|VEKV1EM77B]] $\mathrm{SL}_2(\Z)$ 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|GWKN5VG78A]] of level $N$ are modular forms for these groups.
Most modular forms that arise in arithmetic, such as those attached to [[elliptic curves|427N6S6QQW]], have a congruence subgroup as their symmetry group.

## Explanations

### L1 human

Take the integer matrices of determinant $1$ and reduce their entries modulo $N$.
A congruence subgroup is a subgroup that can be described by conditions on these reductions.
For example, $\Gamma_0(N)$ is the set of matrices whose lower left entry is divisible by $N$.
Such a group is smaller than $\mathrm{SL}_2(\Z)$, so functions symmetric under it are easier to find.