Editing Congruence subgroup (section)

== Congruence subgroups of the modular group ==

The simplest interesting setting in which congruence subgroups can be studied is that of the [[modular group]] {{tmath|1= \mathrm{SL}_2(\Z) }}.<ref>The modular group is usually defined to be the quotient {{tmath|1= \mathrm{PSL}_2(\Z) = \mathrm{SL}_2(\Z) / \{ \pm \operatorname{Id} \} }}, here we will rather use <math>\mathrm{SL}_2(\Z)</math> to make things simpler, but the theory is almost the same.</ref>

=== Principal congruence subgroups ===

If <math>n \geqslant 1</math> is an integer there is a homomorphism <math>\pi_n: \mathrm{SL}_2(\Z) \to \mathrm{SL}_2(\Z /n\Z)</math> induced by the reduction modulo <math>n</math> morphism {{tmath|1= \Z \to \Z / n\Z }}. The ''principal congruence subgroup of level <math>n</math>'' in <math>\Gamma = \mathrm{SL}_2(\Z)</math> is the kernel of {{tmath|1= \pi_n }}, and it is usually denoted {{tmath|1= \Gamma(n) }}. Explicitly it is described as follows:
: <math> \Gamma(n) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\Z) : a, d \equiv 1 \pmod n, \quad b, c \equiv 0 \pmod n \right\} </math>

This definition immediately implies that <math>\Gamma(n)</math> is a [[normal subgroup]] of finite [[Index of a subgroup|index]] in {{tmath|1= \Gamma }}. The [[strong approximation theorem]] (in this case an easy consequence of the [[Chinese remainder theorem]]) implies that <math>\pi_n</math> is surjective, so that the quotient <math>\Gamma /\Gamma(n)</math> is isomorphic to {{tmath|1= \mathrm{SL}_2(\Z/n\Z) }}. Computing the order of this finite group yields the following formula for the index:
: <math> [\Gamma : \Gamma(n)] = n^3 \cdot \prod_{p \mid n} \left( 1 - \frac 1 {p^2} \right) </math>
where the product is taken over all prime numbers dividing {{tmath|1= n }}.

If <math>n \geqslant 3</math> then the restriction of <math>\pi_n</math> to any finite subgroup of <math>\Gamma</math> is injective. This implies the following result: 
: ''If <math>n\geqslant 3</math> then the principal congruence subgroups <math>\Gamma(n)</math> are [[Torsion-free group|torsion-free]].''
The group <math>\Gamma(2)</math> contains <math>-\operatorname{Id}</math> and is not torsion-free. On the other hand, its image in <math>\operatorname{PSL}_2(\Z)</math> is torsion-free, and the quotient of the [[hyperbolic plane]] by this subgroup is a sphere with three cusps.

=== Definition of a congruence subgroup ===

A subgroup <math>H</math> in <math>\Gamma = \mathrm{SL}_2(\Z)</math>  is called a ''congruence subgroup'' if there exists <math>n \geqslant 1</math> such that <math>H</math> contains the principal congruence subgroup {{tmath|1= \Gamma(n) }}. The ''level'' <math>l</math> of <math>H</math> is then the smallest such {{tmath|1= n }}.

From this definition it follows that:
* Congruence subgroups are of finite index in {{tmath|1= \Gamma }}; 
* The congruence subgroups of level <math>\ell</math> are in one-to-one correspondence with the subgroups of {{tmath|1= \operatorname{SL}_2(\Z/\ell\Z ) }}.

=== 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 }}.

=== Properties of congruence subgroups ===

The congruence subgroups of the modular group and the associated Riemann surfaces are distinguished by some particularly nice geometric and topological properties. Here is a sample:
* There are only finitely many congruence covers of the modular surface that have genus zero;<ref>{{cite journal | last1=Long | first1=Darren D. | last2=Maclachlan | first2=Colin | last3=Reid | first3=Alan | title=Arithmetic Fuchsian groups of genus zero | journal=Pure and Applied Math Quarterly 2 | date=2006 | volume=Special issue to celebrate the 60th birthday of Professor J. H. Coates | issue=2 | pages=569&ndash;599| doi=10.4310/PAMQ.2006.v2.n2.a9 | doi-access=free }}</ref> 
* ([[Selberg's 1/4 conjecture|Selberg's 3/16 theorem]]) If <math>f</math> is a nonconstant eigenfunction of the [[Laplace-Beltrami operator]] on a congruence cover of the modular surface with eigenvalue <math>\lambda</math> then {{tmath|1= \lambda \geqslant \tfrac{3}{16} }}.

There is also a collection of distinguished operators called [[Hecke operator]]s on smooth functions on congruence covers, which commute with each other and with the Laplace–Beltrami operator and are diagonalisable in each eigenspace of the latter. Their common eigenfunctions are a fundamental example of [[automorphic form]]s. Other automorphic forms associated to these congruence subgroups are the holomorphic modular forms, which can be interpreted as cohomology classes on the associated Riemann surfaces via the [[Eichler-Shimura isomorphism]].

=== Normalisers of Hecke congruence subgroups ===

The [[normalizer]] <math>\Gamma_0(p)^+</math> of <math>\Gamma_0(p)</math> in <math>\mathrm{SL}_2(\R)</math> has been investigated; one result from the 1970s, due to [[Jean-Pierre Serre]], [[Andrew Ogg]] and [[John G. Thompson]] is that the corresponding [[modular curve]] (the [[Riemann surface]] resulting from taking the quotient of the hyperbolic plane by {{tmath|1= \Gamma_0(p)^+ }}) has [[genus (mathematics)|genus]] zero (i.e., the modular curve is a Riemann sphere) [[if and only if]] {{tmath|1= p }} is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, or 71. When Ogg later heard about the [[monster group]], he noticed that these were precisely the [[prime factor]]s of the size of {{tmath|1= M}}, he wrote up a paper offering a bottle of [[Jack Daniel's]] whiskey to anyone who could explain this fact – this was a starting point for the theory of [[monstrous moonshine]], which explains deep connections between modular function theory and the monster group.