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 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

The simplest interesting setting in which congruence subgroups can be studied is that of the modular group $\mathrm_2(\Z)$.

Principal congruence subgroups

If $n \geqslant 1$ is an integer there is a homomorphism $\pi_n: \mathrm_2(\Z) \to \mathrm_2(\Z /n\Z)$ induced by the reduction modulo $n$ morphism $\Z \to \Z / n\Z$. The ''principal congruence subgroup of level $n$'' in $\Gamma = \mathrm_2(\Z)$ is the kernel of $\pi_n$, and it is usually denoted $\Gamma(n)$. Explicitly it is described as follows:

:$\Gamma(n) = \left\$

This definition immediately implies that $\Gamma(n)$ is a normal subgroup of finite index in $\Gamma$. The strong approximation theorem (in this case an easy consequence of the Chinese remainder theorem) implies that $\pi_n$ is surjective, so that the quotient $\Gamma /\Gamma(n)$ is isomorphic to $\mathrm_2(\Z/n\Z).$ Computing the order of this finite group yields the following formula for the index:

: Gamma : \Gamma(n)= n^3 \cdot \prod_ \left( 1 - \frac 1 \right)

where the product is taken over all prime numbers dividing $n$.

If $n \geqslant 3$ then the restriction of $\pi_n$ to any finite subgroup of $\Gamma$ is injective. This implies the following result:
:''If $n\geqslant 3$ then the principal congruence subgroups $\Gamma(n)$ are torsion-free. ''
The group $\Gamma(2)$ contains $-\operatorname$ and is not torsion-free. On the other hand, its image in $\operatorname_2(\Z)$ 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 $H$ is a subgroup in $\Gamma = \mathrm_2(\Z)$ then it is called a ''congruence subgroup'' if there exists $n \geqslant 1$ such that it contains the principal congruence subgroup $\Gamma(n)$. The ''level'' $l$ of $H$ is then the smallest such $n$.

From this definition it follows that:
*Congruence subgroups are of finite index in $\Gamma$;
*The congruence subgroups of level $\ell$ are in one-to-one correspondence with the subgroups of $\operatorname_2(\Z/\ell\Z ).$

Examples

The subgroups $\Gamma_0(n)$, sometimes called the ''Hecke congruence subgroup'' of level $n$, is defined as the preimage by $\pi_n$ of the group of upper triangular matrices. That is,

:$\Gamma_0(n) = \left\.$

The index is given by the formula:

: Gamma : \Gamma_0(n)= n \cdot \prod_ \left( 1 + \frac 1 p \right)

where the product is taken over all prime numbers dividing $n$. If $p$ is prime then $\Gamma/\Gamma_0(p)$ is in natural bijection with the projective line over the finite field $\mathbb F_p$, and explicit representatives for the (left or right) cosets of $\Gamma_0(p)$ in $\Gamma$ are the following matrices:

:$\operatorname, \begin 1 & 0 \\ 1 & 1 \end, \ldots, \begin 1 & 0 \\ p-1 & 1 \end, \begin 0 & -1 \\ 1 & 0 \end.$

The subgroups $\Gamma_0(n)$ are never torsion-free as they always contain the matrix $-I$. There are infinitely many $n$ such that the image of $\Gamma_0(n)$ in $\mathrm_2(\Z )$ also contains torsion elements.

The subgroups $\Gamma_1(n)$ are the preimage of the subgroup of unipotent matrices:

:$\Gamma_1(n) = \left\.$

Their indices are given by the formula:

: Gamma : \Gamma_1(n)= n^2 \cdot \prod_ \left( 1 - \frac 1 \right)

The ''theta subgroup'' $\Lambda$ is the congruence subgroup of $\Gamma$ defined as the preimage of the cyclic group of order two generated by $\left ( \begin 0 & -1 \\1 & 0 \end \right ) \in \mathrm_2(\Z/2\Z )$. It is of index 3 and is explicitly described by:

:$\Lambda = \left\.$

These subgroups satisfy the following inclusions : $\Gamma(n) \subset \Gamma_1(n) \subset \Gamma_0(n)$, as well as $\Gamma(2) \subset \Lambda$

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 $f$ is a nonconstant eigenfunction of the Laplace-Beltrami operator on a congruence cover of the modular surface with eigenvalue $\lambda$ then $\lambda \geqslant \tfrac.$

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 $\Gamma_0(p)^+$ of $\Gamma_0(p)$ in $\mathrm_2(\R)$ 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 $\Gamma_0(p)^+$) 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

Arithmetic groups

The notion of an arithmetic group is a vast generalisation based upon the fundamental example of $\mathrm_d(\Z)$. In general, to give a definition one needs a semisimple algebraic group $\mathbf G$ defined over $\Q$ and a faithful representation $\rho$, also defined over $\Q,$ from $\mathbf G$ into $\mathrm_d$; then an arithmetic group in $\mathbf G(\Q)$ is any group $\Gamma \subset \mathbf G(\Q)$ which is of finite index in the stabiliser of a finite-index sub-lattice in $\Z^d$.

Congruence subgroups

Let $\Gamma$ be an arithmetic group: for simplicity it is better to suppose that $\Gamma \subset \mathrm_n(\Z)$. As in the case of $\mathrm_2(\Z)$ there are reduction morphisms $\pi_n: \Gamma \to \mathrm_d(\Z/n\Z)$. We can define a principal congruence subgroup of $\Gamma$ to be the kernel of $\pi_n$ (which may a priori depend on the representation $\rho$), and a ''congruence subgroup'' of $\Gamma$ 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 $\pi_n(\Gamma)$, and the level is defined.

Examples

The principal congruence subgroups of $\mathrm_d(\Z )$ are the subgroups $\Gamma(n)$ given by:

:$\Gamma(n) = \left\$

the congruence subgroups then correspond to the subgroups of $\mathrm_d(\Z/n\Z )$.

Another example of arithmetic group is given by the groups $\mathrm_2(O)$ where $O$ is the ring of integers in a number field, for example $O = \Z$sqrt 2/math>. Then if $\mathfrak p$ is a prime ideal dividing a rational prime $p$ the subgroups $\Gamma(\mathfrak p)$ which is the kernel of the reduction map mod $\mathfrak p$ is a congruence subgroup since it contains the principal congruence subgroup defined by reduction modulo $p$.

Yet another arithmetic group is the Siegel modular groups $\mathrm_(\Z),$ defined by:

:$\mathrm_(\Z) = \left\.$

Note that if $g = 1$ then $\mathrm_2(\Z) = \mathrm_2(\Z).$ The ''theta subgroup'' $\Gamma_^$ of $\mathrm_(\Z)$ is the set of all $\left ( \begin A & B \\ C & D \end\right ) \in \mathrm_(\Z)$ such that both $AB^\top$ and $CD^\top$ have even diagonal entries.

Property (τ)

The family of congruence subgroups in a given arithmetic group $\Gamma$ 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 $\Gamma$) is uniformly bounded away from zero, in other words they are a family of expander graphs. There is also a representation-theoretical interpretation: if $\Gamma$ 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 $L^2(G/\Gamma)$ 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 $\mathbf G$ is a $\Q$-group and $S = \$ is a finite set of primes, an $S$-arithmetic subgroup of $\mathbf G(\Q )$ is defined as an arithmetic subgroup but using \Z /p_1,\ldots, 1/p_r instead of $\Z.$ The fundamental example is \operatorname_d(\Z /p_1,\ldots, 1/p_r.

Let $\Gamma_S$ be an $S$-arithmetic group in an algebraic group $\mathbf G \subset \operatorname_d$. If $n$ is an integer not divisible by any prime in $S$, then all primes $p_i$ are invertible modulo $n$ and it follows that there is a morphism $\pi_n: \Gamma_S \to \mathrm_d(\Z/n\Z).$ Thus it is possible to define congruence subgroups in $\Gamma_S$, whose level is always coprime to all primes in $S$.

The congruence subgroup problem

Finite-index subgroups in SL₂(Z)

Congruence subgroups in $\Gamma = \mathrm_2(\Z)$ are finite-index subgroups: it is natural to ask whether they account for all finite-index subgroups in $\Gamma$. 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 $\Gamma /\Gamma'$, where $\Gamma'$ is a normal congruence subgroup, must be a simple group of Lie type (or cyclic), in fact one of the groups $\mathrm_2(\mathbb F_p)$ for a prime $p$. But for every $m$ there are finite-index subgroups $\Gamma' \subset \Gamma$ such that $\Gamma / \Gamma'$ is isomorphic to the alternating group $A_m$ (for example $\Gamma(2)$ 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 $\Gamma(2) \to \Z$; for $m$ large enough the kernel of $\Gamma(2) \to \Z \to \Z/m\Z$ 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 $c_N$ of congruence subgroups in $\Gamma$ of index $N$ satisfies $\log c_N = O \left( (\log N)^2 / \log \log N \right)$. On the other hand, the number $a_N$ of finite index subgroups of index $N$ in $\Gamma$ satisfies $N \log N = O(\log a_N)$, 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

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