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, in which 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.
The existence of congruence subgroups in an arithmetic group 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
Principal congruence subgroups
If is an integer there is a homomorphism induced by the reduction modulo morphism . The principal congruence subgroup of level in is the kernel of , and it is usually denoted . Explicitly it is described as follows:
This definition immediately implies that is a normal subgroup of finite index in . The strong approximation theorem (in this case an easy consequence of the Chinese remainder theorem) implies that is surjective, so that the quotient is isomorphic to Computing the order of this finite group yields the following formula for the index:
where the product is taken over all prime numbers dividing .
If then the restriction of to any finite subgroup of is injective. This implies the following result:
- If then the principal congruence subgroups are torsion-free.
The group contains and is not torsion-free. On the other hand, its image in is torsion-free, and the quotient of the hyperbolic plane by this subgroup is a sphere with three cusps.
Definition of a congruence subgroup
If is a subgroup in then it is called a congruence subgroup if there exists such that it contains the principal congruence subgroup . The level of is then the smallest such .
From this definition it follows that:
- Congruence subgroups are of finite index in ;
- The congruence subgroups of level are in one-to-one correspondence with the subgroups of
The subgroups , sometimes called the Hecke congruence subgroup of level , is defined as the preimage by of the group of upper triangular matrices. That is,
The index is given by the formula:
where the product is taken over all prime numbers dividing . If is prime then is in natural bijection with the projective line over the finite field , and explicit representatives for the (left or right) cosets of in are the following matrices:
The subgroups are never torsion-free as they always contain the matrix . There are infinitely many such that the image of in also contains torsion elements.
The subgroups are the preimage of the subgroup of unipotent matrices:
They are torsion-free as soon as , and their indices are given by the formula:
The theta subgroup is the congruence subgroup of defined as the preimage of the cyclic group of order two generated by . It is of index 3 and is explicitly described by:
These subgroups satisfy the following inclusions : , as well as
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 which have genus zero;
- (Selberg's 3/16 theorem) If is a nonconstant eigenfunction of the Laplace-Beltrami operator on a congruence cover of the modular surface with eigenvalue then
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 of in 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 ) has genus zero (i.e., the modular curve is a Riemann sphere) 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.
In arithmetic groups
The notion of an arithmetic group is a vast generalisation based upon the fundamental example of . In general, to give a definition one needs a semisimple algebraic group defined over and a faithful representation , also defined over from into ; then an arithmetic group in is any group which is of finite index in the stabiliser of a finite-index sub-lattice in .
Let be an arithmetic group: for simplicity it is better to suppose that . As in the case of there are reduction morphisms . We can define a principal congruence subgroup of to be the kernel of (which may a priori depend on the representation ), and a congruence subgroup of to be any subgroup which contains a principal congruence subgroup (a notion which does not depend on a representation). They are subgroups of finite index which correspond to the subgroups of the finite groups , and the level is defined.
The principal congruence subgroups of are the subgroups given by:
the congruence subgroups then correspond to the subgroups of .
Another example of arithmetic group is given by the groups where is the ring of integers in a number field, for example . Then if is a prime ideal dividing a rational prime the subgroups which is the kernel of the reduction map mod is a congruence subgroup since it contains the principal congruence subgroup defined by reduction modulo .
Yet another arithmetic group is the Siegel modular groups defined by:
Note that if then The theta subgroup of is the set of all such that both and have even diagonal entries.
The family of congruence subgroups in a given arithmetic group 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 ) is uniformly bounded away from zero, in other words they are a family of expander graphs. There is also a representation-theoretical interpretation: if 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 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).
In S-arithmetic groups
If is a -group and is a finite set of primes, an -arithmetic subgroup of is defined as an arithmetic subgroup but using instead of The fundamental example is .
Let be an -arithmetic group in an algebraic group . If is an integer not divisible by any prime in , then all primes are invertible modulo and it follows that there is a morphism Thus it is possible to define congruence subgroups in , whose level is always coprime to all primes in .
The congruence subgroup problem
Finite-index subgroups in SL2(Z)
Congruence subgroups in are finite-index subgroups: it is natural to ask whether they account for all finite-index subgroups in . The answer is a resounding "no". This fact was already known to Felix Klein and there are many ways to exhibit many non-congruence finite-index subgroups. For example:
- The simple group in the composition series of a quotient , where is a normal congruence subgroup, must be a simple group of Lie type (or cyclic), in fact one of the groups for a prime . But for every there are finite-index subgroups such that is isomorphic to the alternating group (for example 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 ; for large enough the kernel of 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 of congruence subgroups in of index satisfies . On the other hand, the number of finite index subgroups of index in satisfies , 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, in contrast to the case of , when all finite-index subgroups in are congruence subgroups. The solution by Bass–Milnor–Serre involved an aspect of algebraic number theory linked to K-theory. On the other hand, the work of Serre on over number fields shows that in some cases the answer to the naïve question is "no" while a slight relaxation of the problem has a positive answer.
This new problem is better stated in terms of certain compact topological groups associated to an arithmetic group . There is a topology on for which a base of neighbourhoods of the trivial subgroup is the set of subgroups of finite index (the profinite topology); and there is another topology defined in the same way using only congruence subgroups. The profinite topology gives rise to a completion of , while the "congruence" topology gives rise to another completion . Both are profinite groups and there is a natural surjective morphism (intuitively, there are fewer conditions for a Cauchy sequence to comply with in the congruence topology than in the profinite topology). The congruence kernel is the kernel of this morphism, and the congruence subgroup problem stated above amounts to whether is trivial. The weakening of the conclusion then leads to the following problem.
- Congruence subgroup problem: Is the congruence kernel finite?
When the problem has a positive solution one says that has the congruence subgroup property. A conjecture generally attributed to Serre states that an irreducible arithmetic lattice in a semisimple Lie group has the congruence subgroup property if and only if the real rank of is at least 2; for example, lattices in should always have the property.
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 , the unitary groups and the groups (the isometry groups of a sesquilinear form over the Hamilton quaternions), plus the exceptional group (see List of simple Lie groups). The current status of the congruence subgroup problem is as follows:
- It is known to have a negative solution (confirming the conjecture) for all groups with . The proof uses the same argument as 2. in the case of : in the general case it is much harder to construct a surjection to the proof is not at all uniform for all cases and fails for some lattices in dimension 7 due to the phenomenon of triality. In dimensions 2 and 3 and for some lattices in higher dimensions argument 1 and 3 also apply.
- It is known for many lattices in , but not all (again using a generalisation of argument 2).
- It is completely open in all remaining cases.
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:
- Any non-anisotropic group (this includes the cases dealt with by Bass–Milnor–Serre, as well as is , and many others);
- Any group of type not (for example all anisotropic forms of symplectic or orthogonal groups of real rank );
- Outer forms of type , for example unitary groups.
The case of inner forms of type is still open. The algebraic groups involved are those associated to the unit groups in central simple division algebras; for example the congruence subgroup property is not known to hold for lattices in or with compact quotient.
Congruence groups and adèle groups
where the product is over all primes and is the field of p-adic numbers. Given any algebraic group over the adelic algebraic group is well-defined. It can be endowed with a canonical topology, which in the case where is a linear algebraic group is the topology as a subset of . The finite adèles are the restricted product of all non-archimedean completions (all p-adic fields).
If is an arithmetic group then its congruence subgroups are characterised by the following property: is a congruence subgroup if and only if its closure is a compact-open subgroup (compactness is automatic) and . In general the group is equal to the congruence closure of in and the congruence topology on is the induced topology as a subgroup of , in particular the congruence completion is its closure in that group. These remarks are also valid for S-arithmetic subgroups, replacing the ring of finite adèles with the restricted product over all primes not in S.
More generally one can define what it means for a subgroup to be a congruence subgroup without explicit reference to a fixed arithmetic subgroup, by asking that it be equal to its congruence closure Thus it becomes possible to study all congruence subgroups at once by looking at the discrete subgroup This is especially convenient in the theory of automorphic forms: for example all modern treatments of the Arthur-Selberg trace formula are done in this adélic setting.
- The modular group is usually defined to be the quotient here we will rather use to make things simpler, but the theory is almost the same.
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