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.

In general given a group such as the special linear group SL(n, Z) we can reduce the entries to modular arithmetic in Z/NZ 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; in 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 then 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 Γ₀(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.

It is not in general true, but is an interesting question to understand, whether all subgroups of finite index are congruence subgroups, in these examples. This question was resolved in the 1960s.

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 pro-finite 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 Bass, Serre and 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 look at many G/Γ as a whole system - is now normal in abstract treatments.

Congruence subgroups of the modular group

See also modular curve

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

For a given positive integer r, the modular group Γ₀(r) is defined as follows:

<math>\Gamma_0(r) := \left\{\begin{bmatrix} a&b\\c&d \end{bmatrix} \in \Gamma : c\equiv 0\mod r\right\}.<math>

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

<math>R_\Gamma \cup \bigcup_{k=0}^{p-1} ST^k(R_\Gamma)<math>

(where Sτ = -1/τ and Tτ = τ+1) is a fundamental region of Γ₀(r).

The modular group Λ (also called the theta subgroup) is another subgroup of the modular group Γ. It can be characterized as the set of linear Möbius transformations w that satisfy

<math>w(t) = \frac{at + b}{ct + d}<math>

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).