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 2×2 integer matrices of determinant 1, such that the off-diagonal entries are even. 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' and can define reduction maps modulo an integer.
Contents
- Congruence subgroups of the modular group
- Principal congruence subgroups
- Definition of a congruence subgroup
- Examples
- Properties of congruence subgroups
- Normalisers of Hecke congruence subgroups
- Arithmetic groups
- Congruence subgroups
- Property
- Congruence subgroups in S arithmetic groups
- Finite index subgroups in SL2Z
- The congruence kernel
- Negative solutions
- Positive solutions
- Congruence groups and adle groups
- References
The existence of congruence subgroups in an arithmetic groups provides it with a wealth of subgroups, in particular it shows that the group is residually finite. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite index are essentially congruence subgroups.
Congruence subgroups of 2×2 matrices are fundamental objects in the classical theory of modular forms; the modern theory of automorphic forms makes a similar use of congruence subgroups in more general arithmetic groups.
Congruence subgroups of the modular group
The simplest interesting setting in which congruence subgroups can be studied is that of the modular group
Principal congruence subgroups
If
This definition immediately implies that
where the product is taken over all prime numbers dividing
If
The groups
Definition of a congruence subgroup
If
From this definition it follows that:
Examples
The subgroups
The index is given by the formula:
where the product is taken over all prime numbers dividing
The subgroups
The subgroups
They are torsion-free as soon as
The theta subgroup
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 is also a collection of distinguished operators called Hecke operators 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 forms. 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 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.
Arithmetic groups
The notion of an arithmetic group is a vast generalisation based upon the fundamental example of
Congruence subgroups
Let
Examples
The principal congruence subgroups of
the congruence subgroups then correspond to the subgroups of
Another example of arithmetic group is given by the groups
Yet another arithmetic group is the Siegel modular groups
Note that if
Property (τ)
The family of congruence subgroups in a given arithmetic group Γ always has property (τ) of Lubotzky–Zimmer. This can be taken to mean that the Cheeger constant of the family of their Schreier coset graphs (with respect to a fixed generating set for Γ) is uniformly bounded away from zero, in other words they are a family of expander graphs. There is also a representation-theoretical interpretation: if Γ is a lattice in a Lie group G then property (τ) is equivalent to the non-trivial unitary representations of G occurring in the spaces L2(G/Γ) being bounded away from the trivial representation (in the Fell topology on the unitary dual of G). Property (τ) is a weakening of Kazhdan's property (T) which implies that the family of all finite-index subgroups has property (τ).
Congruence subgroups in S-arithmetic groups
If
Let
Finite-index subgroups in SL2(Z)
Congruence subgroups in
- The simple group in the composition series of a quotient
Γ / Γ ′ Γ ′ S L 2 ( F p ) for a primep . But for everym there are finite-index subgroupsΓ ′ ⊂ Γ such thatΓ / Γ ′ A m Γ ( 2 ) surjects on any group with two generators, in particular on all alternating groups, and the kernels of these morphisms give an example). These groups thus must be non-congruence. - There is a surjection
Γ ( 2 ) → Z ; form large enough the kernel ofΓ ( 2 ) → Z → Z / m Z must be non-congruence (one way to see this is that the Cheeger constant of the Schreier graph goes to 0; there is also a simple algebraic proof in the spirit of the previous item). - The number
c N Γ of indexN satisfieslog c N = O ( ( log N ) 2 / log log N ) . On the other hand, the numbera N N inΓ satisfiesN log N = O ( log a N ) , so most subgroups of finite index must be non-congruence.
The congruence kernel
One can ask the same question for any arithmetic group as for the modular group:
Naïve Congruence subgroup problem: Given an arithmetic group, are all of its finite-index subgroups congruence subgroups?This problem can have a positive solution: its origin is in the work of Hyman Bass, Jean-Pierre Serre and John Milnor, and Jens Mennicke, who proved that all finite-index subgroups in
This new problem is better stated in terms of certain compact topological groups associated to an arithmetic group
When the problem has a positive solution one says that
Negative solutions
Serre's conjecture states that a lattice in a Lie group of rank one should not have the congruence subgroup property. There are three families of such groups: the orthogonal groups
Positive solutions
In many situations where the congruence subgroup problem is expected to have a positive solution it has been proven that this is indeed the case. Here is a list of algebraic groups such that the congruence subgroup property is known to hold for the associated arithmetic lattices, in case the rank of the associated Lie group (or more generally the sum of the rank of the real and p-adic factors in the case of S-arithmetic groups) is at least 2:
The case of inner forms of type
Congruence groups and adèle groups
The ring of adeles
If
More generally one can define what it means for a subgroup