# Congruence subgroup

In mathematics, a **congruence subgroup** of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are *even*.

An important class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the special linear group SL(n, **Z**) we can reduce the entries to modular arithmetic in **Z**/N**Z** for any N >1, which gives a homomorphism

*SL*(*n*,**Z**) →*SL*(*n*,**Z**/*N*·**Z**)

of groups. The kernel of this reduction map is an example of a congruence subgroup – the condition is that the diagonal entries are congruent to 1 mod *N,* and the off-diagonal entries be congruent to 0 mod *N* (divisible by *N*), and is known as a **Template:Visible anchor**, Γ(*N*). Formally a congruence subgroup is one that contains Γ(*N*) for some *N*,^{[1]} and the least such *N* is the *level* or *Stufe* of the subgroup.

In the case *n=2* we are talking then about a subgroup of the modular group (up to the quotient by {I,-I} taking us to the corresponding projective group): the kernel of reduction is called Γ(N) and plays a big role in the theory of modular forms. Further, we may take the inverse image of any subgroup (not just {e}) and get a congruence subgroup: the subgroups Γ_{0}(N) important in modular form theory are defined in this way, from the subgroup of mod *N* *2x2* matrices with 1 on the diagonal and 0 below it.

More generally, the notion of **congruence subgroup** can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' respected by the subgroup, and so some general idea of what 'congruence' means.

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## Congruence subgroups and topological groups

Are all subgroups of finite index actually congruence subgroups? This is not in general true, and *non-congruence subgroups* exist. It is however an interesting question to understand when these examples are possible. This problem about the classical groups was resolved by Template:Harvtxt. .

It can be posed in topological terms: if Γ is some arithmetic group, there is a topology on Γ for which a base of neighbourhoods of {e} is the set of subgroups of finite index; and there is another topology defined in the same way using only congruence subgroups. We can ask whether those are the same topologies; equivalently, if they give rise to the same completions. The subgroups of finite index give rise to the completion of Γ as a profinite group. If there are essentially fewer congruence subgroups, the corresponding completion of Γ can be bigger (intuitively, there are fewer conditions for a Cauchy sequence to comply with). Therefore the problem can be posed as a relationship of two compact topological groups, with the question reduced to calculation of a possible kernel. The solution by Hyman Bass, Jean-Pierre Serre and John Milnor involved an aspect of algebraic number theory linked to K-theory.

The use of adele methods for automorphic representations (for example in the Langlands program) implicitly uses that kind of completion with respect to a congruence subgroup topology - for the reason that then all congruence subgroups can then be treated within a single group representation. This approach - using a group G(**A**) and its single quotient G(**A**)/G(**Q**) rather than looking at many G/Γ as a whole system - is now normal in abstract treatments.

## Congruence subgroups of the modular group

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Detailed information about the congruence subgroups of the modular group Γ has proved basic in much research, in number theory, and in other areas such as monstrous moonshine.

### Modular group Γ(*r*)

For a given positive integer *r*, the modular group **Γ( r)** is defined as follows:

^{[2]}

### Modular group Γ_{1}(*r*)

For a given positive integer *r*, the modular group **Γ _{1}(r)** is defined as follows:

^{[2]}

### Modular group Γ_{0}(*r*)

For a given positive integer *r*, the modular group **Γ _{0}(r)** is defined as follows:

^{[2]}

It can be shown that for a prime number *p*, the set

(where *S*τ = −1/τ and *T*τ = τ + 1) is a fundamental region of Γ_{0}(*r*).

The normalizer Γ_{0}(*p*)^{+} of Γ_{0}(*p*) in *SL*(2,**R**) 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 Γ_{0}(*p*)^{+}) has genus zero (the modular curve is an elliptic curve) if and only if *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 factors of the size of *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.

### Modular group Λ

The **modular group Λ** is another subgroup of the modular group Γ. It can be characterized as the set of linear Möbius transformations *w* that satisfy

with *a* and *d* being odd and *b* and *c* being even. That is, it is the congruence subgroup that is the kernel of reduction modulo 2, otherwise known as Γ(2).

## Congruence subgroups of the Siegel modular group

The **Siegel modular group** Sp(n, **Z**) is the group of all 2n by 2n matrices with integer entries defined as follows:^{[3]}

where denotes the transpose.

### Theta subgroup

The **theta subgroup** of Sp(n, **Z**) is the set of all in Sp(n, **Z**) such that both and have even diagonal entries.^{[4]}

## References

- ↑ Lang (1976) p.26
- ↑
^{2.0}^{2.1}^{2.2}Lang (1976) p.29 - ↑ Birman, Joan S. "On Siegel's modular group." Mathematische Annalen 191.1 (1971): 59-68.
- ↑ Richter, Olav. "Theta functions of indefinite quadratic forms over real number fields." Proceedings of the American Mathematical Society 128.3 (2000): 701-708.

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