Editing Congruence subgroup (section)

=== Examples ===

The subgroup {{tmath|1= \Gamma_0(n) }}, sometimes called the ''Hecke congruence subgroup'' of level {{tmath|1= n }}, is defined as the preimage by <math>\pi_n</math> of the group of upper triangular matrices. That is,
: <math> \Gamma_0(n) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma : c \equiv 0 \pmod n \right\}. </math>

The index is given by the formula:
: <math> [\Gamma : \Gamma_0(n)] = n \cdot \prod_{p | n} \left( 1 + \frac 1 p \right) </math>
where the product is taken over all prime numbers dividing {{tmath|1= n }}. If <math>p</math> is prime then <math>\Gamma/\Gamma_0(p)</math> is in natural bijection with the [[projective line]] over the finite field {{tmath|1= \mathbb F_p }}, and explicit representatives for the (left or right) cosets of <math>\Gamma_0(p)</math> in <math>\Gamma</math> are the following matrices:
: <math> \operatorname{Id}, \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \ldots, \begin{pmatrix} 1 & 0 \\ p-1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}. </math>

The subgroups <math>\Gamma_0(n)</math> are never torsion-free as they always contain the matrix {{tmath|1= -I }}. There are infinitely many <math> n </math> such that the image of <math>\Gamma_0(n)</math> in <math>\mathrm{PSL}_2(\Z )</math> also contains torsion elements.

The subgroup <math>\Gamma_1(n)</math> is the preimage of the subgroup of unipotent matrices:
: <math> \Gamma_1(n) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma : a, d \equiv 1 \pmod n, c \equiv 0 \pmod n \right\}. </math>

Their indices are given by the formula:
: <math> [\Gamma : \Gamma_1(n)] = n^2 \cdot \prod_{p | n} \left( 1 - \frac 1 {p^2} \right) </math>

The ''theta subgroup'' <math>\Lambda</math> is the congruence subgroup of <math>\Gamma</math> defined as the preimage of the cyclic group of order two generated by <math>\left ( \begin{smallmatrix} 0 & -1 \\1 & 0 \end{smallmatrix} \right ) \in \mathrm{SL}_2(\Z/2\Z )</math>. It is of index 3 and is explicitly described by:<ref>{{cite book | last=Eichler | first=Martin | title=Introduction to the Theory of Algebraic Numbers and Functions | url=https://archive.org/details/introductiontoth0000eich | url-access=registration | publisher=Academic Press | year=1966 | pages=[https://archive.org/details/introductiontoth0000eich/page/36 36]&ndash;39}}</ref>
: <math> \Lambda = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma : ac \equiv 0 \pmod 2, bd \equiv 0 \pmod 2 \right\}.</math>

These subgroups satisfy the following inclusions: {{tmath|1= \Gamma(n) \subset \Gamma_1(n) \subset \Gamma_0(n) }}, as well as {{tmath|1= \Gamma(2) \subset \Lambda }}.