A very simple example is the subgroup of 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.
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.
The simplest interesting setting in which congruence subgroups can be studied is that of the modular group
Computing the order of this finite group yields the following formula for the index: where the product is taken over all prime numbers dividing
is torsion-free, and the quotient of the hyperbolic plane by this subgroup is a sphere with three cusps.
is in natural bijection with the projective line over the finite field
defined as the preimage of the cyclic group of order two generated by
The congruence subgroups of the modular group and the associated Riemann surfaces are distinguished by some particularly nice geometric and topological properties.
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
When Ogg later heard about the monster group, he noticed that these were precisely the prime factors of the size of
, 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.
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
[5] 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.
then property (τ) is equivalent to the non-trivial unitary representations of
being bounded away from the trivial representation (in the Fell topology on the unitary dual of
This fact was already known to Felix Klein and there are many ways to exhibit many non-congruence finite-index subgroups.
For example: One can ask the same question for any arithmetic group as for the modular group: 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
The solution by Bass–Milnor–Serre involved an aspect of algebraic number theory linked to K-theory.
is the kernel of this morphism, and the congruence subgroup problem stated above amounts to whether
A conjecture generally attributed to Serre states that an irreducible arithmetic lattice in a semisimple Lie group
Serre's conjecture states that a lattice in a Lie group of rank one should not have the congruence subgroup property.
The current status of the congruence subgroup problem is as follows: 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
are those associated to the unit groups in central simple division algebras; for example the congruence subgroup property is not known for lattices in
are the restricted product of all non-archimedean completions (all p-adic fields).
-arithmetic subgroups, replacing the ring of finite adèles with the restricted product over all primes not in
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.