Congruence subgroup

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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/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 – 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 principal congruence subgroup, 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 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.

Congruence subgroups and topological groups[edit]

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 Bass, Milnor & Serre (1967). .

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[edit]

See also: modular curve

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)[edit]

For a given positive integer r, the modular group Γ(r) is defined as follows:[2]

\Gamma(r) := \left\{\begin{bmatrix} a&b\\c&d \end{bmatrix} \in \Gamma : a\equiv d\equiv \pm 1,~b\equiv c\equiv 0\mod r\right\}.

Modular group Γ1(r)[edit]

For a given positive integer r, the modular group Γ1(r) is defined as follows:[2]

\Gamma_1(r) := \left\{\begin{bmatrix} a&b\\c&d \end{bmatrix} \in \Gamma :  a\equiv d\equiv 1,~c\equiv 0\mod r\right\}.

Modular group Γ0(r)[edit]

For a given positive integer r, the modular group Γ0(r) is defined as follows:[2]

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

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

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

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

The normalizer Γ0(p)+ of Γ0(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 Γ0(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[edit]

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

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

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).

References[edit]

  1. ^ Lang (1976) p.26
  2. ^ a b c Lang (1976) p.29