Congruence subgroup

In arithmetic groups

Arithmetic groups

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

Congruence subgroups

Let ${\displaystyle \mathrm{\Gamma }}$ be an arithmetic group: for simplicity it is better to suppose that ⁠${\displaystyle \mathrm{\Gamma }\subset {\mathrm{G}\mathrm{L}}_n(\mathbb{Z})}$⁠. As in the case of ${\displaystyle {\mathrm{S}\mathrm{L}}_2(\mathbb{Z})}$ there are reduction morphisms ⁠${\displaystyle {\pi }_n:\mathrm{\Gamma }\to {\mathrm{G}\mathrm{L}}_d(\mathbb{Z}/n\mathbb{Z})}$⁠. We can define a principal congruence subgroup of ${\displaystyle \mathrm{\Gamma }}$ to be the kernel of ${\displaystyle {\pi }_n}$ (which may a priori depend on the representation ⁠${\displaystyle \rho }$⁠), and a congruence subgroup of ${\displaystyle \mathrm{\Gamma }}$ to be any subgroup that contains a principal congruence subgroup (a notion that does not depend on a representation). They are subgroups of finite index that correspond to the subgroups of the finite groups ⁠${\displaystyle {\pi }_n(\mathrm{\Gamma })}$⁠, and the level is defined.

Examples

The principal congruence subgroups of ${\displaystyle {\mathrm{S}\mathrm{L}}_d(\mathbb{Z})}$ are the subgroups ${\displaystyle \mathrm{\Gamma }(n)}$ given by:

${\displaystyle \mathrm{\Gamma }(n)=\left\{(a_{ij})\in {\mathrm{S}\mathrm{L}}_d(\mathbb{Z}):\mathrm{\forall }i{a}_{ii}\equiv 1(\mathrm{mod}n),\mathrm{\forall }i\ne j{a}_{ij}\equiv 0(\mathrm{mod}n)\right\}}$

the congruence subgroups then correspond to the subgroups of ${\displaystyle {\mathrm{S}\mathrm{L}}_d(\mathbb{Z}/n\mathbb{Z})}$.

Another example of arithmetic group is given by the groups ${\displaystyle {\mathrm{S}\mathrm{L}}_2(O)}$ where ${\displaystyle O}$ is the ring of integers in a number field, for example ⁠${\displaystyle O=\mathbb{Z}[\sqrt{2}]}$⁠. Then if ${\displaystyle \mathfrak{p}}$ is a prime ideal dividing a rational prime ${\displaystyle p}$ the subgroups ${\displaystyle \mathrm{\Gamma }(\mathfrak{p})}$ that is the kernel of the reduction map mod ${\displaystyle \mathfrak{p}}$ is a congruence subgroup since it contains the principal congruence subgroup defined by reduction modulo ⁠${\displaystyle p}$⁠.

Yet another arithmetic group is the Siegel modular groups ⁠${\displaystyle {\mathrm{S}\mathrm{p}}_{2g}(\mathbb{Z})}$⁠, defined by:

${\displaystyle {\mathrm{S}\mathrm{p}}_{2g}(\mathbb{Z})=\left\{\gamma \in {\mathrm{G}\mathrm{L}}_{2g}(\mathbb{Z}):{\gamma }^{\mathrm{T}}\left(\begin{array}{cc}0& I_g\\ -I_g& 0\end{array}\right)\gamma =\left(\begin{array}{cc}0& I_g\\ -I_g& 0\end{array}\right)\right\}.}$

Note that if ${\displaystyle g=1}$ then ⁠${\displaystyle {\mathrm{S}\mathrm{p}}_2(\mathbb{Z})={\mathrm{S}\mathrm{L}}_2(\mathbb{Z})}$⁠. The theta subgroup ${\displaystyle {\mathrm{\Gamma }}_{\vartheta }^{(n)}}$ of ${\displaystyle {\mathrm{S}\mathrm{p}}_{2g}(\mathbb{Z})}$ is the set of all ${\displaystyle \left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in {\mathrm{S}\mathrm{p}}_{2g}(\mathbb{Z})}$ such that both ${\displaystyle A{B}^{\mathrm{\top }}}$ and ${\displaystyle C{D}^{\mathrm{\top }}}$ have even diagonal entries.

Property (τ)

The family of congruence subgroups in a given arithmetic group ${\displaystyle \mathrm{\Gamma }}$ 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 ⁠${\displaystyle \mathrm{\Gamma }}$⁠) is uniformly bounded away from zero, in other words they are a family of expander graphs. There is also a representation-theoretical interpretation: if ${\displaystyle \mathrm{\Gamma }}$ is a lattice in a Lie group ⁠${\displaystyle G}$⁠ then property (τ) is equivalent to the non-trivial unitary representations of ⁠${\displaystyle G}$⁠ occurring in the spaces ${\displaystyle L^2(G/\mathrm{\Gamma })}$ being bounded away from the trivial representation (in the Fell topology on the unitary dual of ⁠${\displaystyle G}$⁠). 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 ${\displaystyle \mathbf{G}}$ is a ${\displaystyle \mathbb{Q}}$-group and ${\displaystyle S=\{p_1,\dots ,p_r\}}$ is a finite set of primes, an ${\displaystyle S}$-arithmetic subgroup of ${\displaystyle \mathbf{G}(\mathbb{Q})}$ is defined as an arithmetic subgroup but using ${\displaystyle \mathbb{Z}[1/p_1,\dots ,1/p_r])}$ instead of ⁠${\displaystyle \mathbb{Z}}$⁠. The fundamental example is ⁠${\displaystyle {\mathrm{SL}}_d(\mathbb{Z}[1/p_1,\dots ,1/p_r])}$⁠.

Let ${\displaystyle {\mathrm{\Gamma }}_S}$ be an ${\displaystyle S}$-arithmetic group in an algebraic group ⁠${\displaystyle \mathbf{G}\subset {\mathrm{GL}}_d}$⁠. If ${\displaystyle n}$ is an integer not divisible by any prime in ⁠${\displaystyle S}$⁠, then all primes ${\displaystyle p_i}$ are invertible modulo ${\displaystyle n}$ and it follows that there is a morphism ⁠${\displaystyle {\pi }_n:{\mathrm{\Gamma }}_S\to {\mathrm{G}\mathrm{L}}_d(\mathbb{Z}/n\mathbb{Z})}$⁠. Thus it is possible to define congruence subgroups in ⁠${\displaystyle {\mathrm{\Gamma }}_S}$⁠, whose level is always coprime to all primes in ⁠${\displaystyle S}$⁠.

The congruence subgroup problem

Finite-index subgroups in SL2(Z)

Congruence subgroups in ${\displaystyle \mathrm{\Gamma }={\mathrm{S}\mathrm{L}}_2(\mathbb{Z})}$ are finite-index subgroups: it is natural to ask whether they account for all finite-index subgroups in ⁠${\displaystyle \mathrm{\Gamma }}$⁠. 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:

1. The simple group in the composition series of a quotient ⁠${\displaystyle \mathrm{\Gamma }/{\mathrm{\Gamma }}^{\prime }}$⁠, where ${\displaystyle {\mathrm{\Gamma }}^{\prime }}$ is a normal congruence subgroup, must be a simple group of Lie type (or cyclic), in fact one of the groups ${\displaystyle {\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)}$ for a prime ⁠${\displaystyle p}$⁠. But for every ${\displaystyle m}$ there are finite-index subgroups ${\displaystyle {\mathrm{\Gamma }}^{\prime }\subset \mathrm{\Gamma }}$ such that ${\displaystyle \mathrm{\Gamma }/{\mathrm{\Gamma }}^{\prime }}$ is isomorphic to the alternating group ${\displaystyle A_m}$ (for example ${\displaystyle \mathrm{\Gamma }(2)}$ 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.
2. There is a surjection ⁠${\displaystyle \mathrm{\Gamma }(2)\to \mathbb{Z}}$⁠; for ${\displaystyle m}$ large enough the kernel of ${\displaystyle \mathrm{\Gamma }(2)\to \mathbb{Z}\to \mathbb{Z}/m\mathbb{Z}}$ 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).
3. The number ${\displaystyle c_N}$ of congruence subgroups in ${\displaystyle \mathrm{\Gamma }}$ of index ${\displaystyle N}$ satisfies ⁠${\displaystyle \log c_N=O\left((\log N{)}^2/\log \log N\right)}$⁠. On the other hand, the number ${\displaystyle a_N}$ of finite index subgroups of index ${\displaystyle N}$ in ${\displaystyle \mathrm{\Gamma }}$ satisfies ⁠${\displaystyle N\log N=O(\log a_N)}$⁠, so most subgroups of finite index must be non-congruence.

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 ${\displaystyle {\mathrm{S}\mathrm{L}}_2(\mathbb{Z})}$, when ${\displaystyle n⩾3}$ all finite-index subgroups in ${\displaystyle {\mathrm{S}\mathrm{L}}_n(\mathbb{Z})}$ 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 ${\displaystyle {\mathrm{S}\mathrm{L}}_2}$ 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 ⁠${\displaystyle \mathrm{\Gamma }}$⁠. There is a topology on ${\displaystyle \mathrm{\Gamma }}$ 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 ⁠${\displaystyle \hat{\mathrm{\Gamma }}}$⁠ of ⁠${\displaystyle \mathrm{\Gamma }}$⁠, while the "congruence" topology gives rise to another completion ⁠${\displaystyle \overline{\mathrm{\Gamma }}}$⁠. Both are profinite groups and there is a natural surjective morphism ${\displaystyle \hat{\mathrm{\Gamma }}\to \overline{\mathrm{\Gamma }}}$ (intuitively, there are fewer conditions for a Cauchy sequence to comply with in the congruence topology than in the profinite topology). The congruence kernel ${\displaystyle C(\mathrm{\Gamma })}$ is the kernel of this morphism, and the congruence subgroup problem stated above amounts to whether ${\displaystyle C(\mathrm{\Gamma })}$ is trivial. The weakening of the conclusion then leads to the following problem.

Congruence subgroup problem: Is the congruence kernel ${\displaystyle C(\mathrm{\Gamma })}$ finite?

When the problem has a positive solution one says that ${\displaystyle \mathrm{\Gamma }}$ has the congruence subgroup property. A conjecture generally attributed to Serre states that an irreducible arithmetic lattice in a semisimple Lie group ${\displaystyle G}$ has the congruence subgroup property if and only if the real rank of ${\displaystyle G}$ is at least 2; for example, lattices in ${\displaystyle {\mathrm{S}\mathrm{L}}_3(\mathbb{R})}$ should always have the property.

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 ⁠${\displaystyle \mathrm{S}\mathrm{O}(d,1),d⩾2}$⁠, the unitary groups ${\displaystyle \mathrm{S}\mathrm{U}(d,1),d⩾2}$ and the groups ${\displaystyle \mathrm{S}\mathrm{p}(d,1),d⩾2}$ (the isometry groups of a sesquilinear form over the Hamilton quaternions), plus the exceptional group ${\displaystyle F_4^{-20}}$ (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 ${\displaystyle \mathrm{S}\mathrm{O}(d,1)}$ with ⁠${\displaystyle d\ne 7}$⁠. The proof uses the same argument as 2. in the case of ${\displaystyle {\mathrm{S}\mathrm{L}}_2(\mathbb{Z})}$: in the general case it is much harder to construct a surjection to ⁠${\displaystyle \mathbb{Z}}$⁠, 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 ${\displaystyle \mathrm{S}\mathrm{U}(d,1)}$, but not all (again using a generalisation of argument 2).
• It is completely open in all remaining cases.

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 ⁠${\displaystyle p}$⁠-adic factors in the case of ⁠${\displaystyle S}$⁠-arithmetic groups) is at least 2:

• Any non-anisotropic group (this includes the cases dealt with by Bass–Milnor–Serre, as well as ${\displaystyle \mathrm{S}\mathrm{O}(p,q)}$ is ⁠${\displaystyle min(p,q)>1}$⁠, and many others);
• Any group of type not ${\displaystyle A_n}$ (for example all anisotropic forms of symplectic or orthogonal groups of real rank ⁠${\displaystyle ⩾2}$⁠);
• Unitary groups of hermitian forms.

The cases of inner and outer forms of type ${\displaystyle A_n}$ are still open. The algebraic groups in the case of inner forms of type ${\displaystyle A_n}$ are those associated to the unit groups in central simple division algebras; for example the congruence subgroup property is not known for lattices in ${\displaystyle {\mathrm{S}\mathrm{L}}_3(\mathbb{R})}$ or ${\displaystyle {\mathrm{S}\mathrm{L}}_2(\mathbb{R})\times {\mathrm{S}\mathrm{L}}_2(\mathbb{R})}$ with compact quotient.

Congruence groups and adèle groups

The ring of adeles ${\displaystyle \mathbb{A}}$ is the restricted product of all completions of ⁠${\displaystyle \mathbb{Q}}$⁠, i.e.

${\displaystyle \mathbb{A}=\mathbb{R}\times {\prod _p}^{\prime }{\mathbb{Q}}_p}$

where the product is over the set ${\displaystyle \mathcal{P}}$ of all primes, ${\displaystyle {\mathbb{Q}}_p}$ is the field of p-adic numbers and an element ${\displaystyle (x,(x_p{)}_{p\in \mathcal{P}})}$ belongs to the restricted product if and only if for almost all primes ⁠${\displaystyle p}$⁠, ${\displaystyle x_p}$ belongs to the subring ${\displaystyle {\mathbb{Z}}_p}$ of p-adic integers.

Given any algebraic group ${\displaystyle \mathbf{G}}$ over ${\displaystyle \mathbb{Q}}$ the adelic algebraic group ${\displaystyle \mathbf{G}(\mathbb{A})}$ is well-defined. It can be endowed with a canonical topology, which in the case where ${\displaystyle \mathbf{G}}$ is a linear algebraic group is the topology as a subset of ⁠${\displaystyle {\mathbb{A}}^m}$⁠. The finite adèles ${\displaystyle {\mathbb{A}}_f}$ are the restricted product of all non-archimedean completions (all p-adic fields).

If ${\displaystyle \mathrm{\Gamma }\subset \mathbf{G}(\mathbb{Q})}$ is an arithmetic group then its congruence subgroups are characterised by the following property: ${\displaystyle H\subset \mathrm{\Gamma }}$ is a congruence subgroup if and only if its closure ${\displaystyle \overline{H}\subset \mathbf{G}({\mathbb{A}}_f)}$ is a compact-open subgroup (compactness is automatic) and ⁠${\displaystyle H=\mathrm{\Gamma }\cap \overline{H}}$⁠. In general the group ${\displaystyle \mathrm{\Gamma }\cap \overline{H}}$ is equal to the congruence closure of ${\displaystyle H}$ in ⁠${\displaystyle \mathrm{\Gamma }}$⁠, and the congruence topology on ${\displaystyle \mathrm{\Gamma }}$ is the induced topology as a subgroup of ⁠${\displaystyle \mathbf{G}({\mathbb{A}}_f)}$⁠, in particular the congruence completion ${\displaystyle \overline{\mathrm{\Gamma }}}$ is its closure in that group. These remarks are also valid for ⁠${\displaystyle S}$⁠-arithmetic subgroups, replacing the ring of finite adèles with the restricted product over all primes not in ⁠${\displaystyle S}$⁠.

More generally one can define what it means for a subgroup ${\displaystyle \mathrm{\Gamma }\subset \mathbf{G}(\mathbb{Q})}$ to be a congruence subgroup without explicit reference to a fixed arithmetic subgroup, by asking that it be equal to its congruence closure ⁠${\displaystyle \overline{\mathrm{\Gamma }}\cap \mathbf{G}(\mathbb{Q})}$⁠. Thus it becomes possible to study all congruence subgroups at once by looking at the discrete subgroup ⁠${\displaystyle \mathbf{G}(\mathbb{Q})\subset \mathbf{G}(\mathbb{A})}$⁠. 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.