Editing Congruence subgroup (section)

== In arithmetic groups ==

=== Arithmetic groups ===
{{Main article | Arithmetic group}}

The notion of an arithmetic group is a vast generalisation based upon the fundamental example of {{tmath|1= \mathrm{SL}_d(\Z) }}. In general, to give a definition one needs a [[semisimple algebraic group]] <math>\mathbf G</math> defined over <math>\Q</math> and a faithful representation {{tmath|1= \rho }}, also defined over {{tmath|1= \Q }}, from <math>\mathbf G</math> into {{tmath|1= \mathrm{GL}_d }}; then an arithmetic group in <math>\mathbf G(\Q)</math> is any group <math>\Gamma \subset \mathbf G(\Q)</math> that is of finite index in the stabiliser of a finite-index sub-lattice in {{tmath|1= \Z^d }}.

=== Congruence subgroups ===

Let <math>\Gamma</math> be an arithmetic group: for simplicity it is better to suppose that {{tmath|1= \Gamma \subset \mathrm{GL}_n(\Z) }}. As in the case of <math>\mathrm{SL}_2(\Z)</math> there are reduction morphisms {{tmath|1= \pi_n: \Gamma \to \mathrm{GL}_d(\Z/n\Z) }}. We can define a principal congruence subgroup of <math>\Gamma</math> to be the kernel of <math>\pi_n</math> (which may a priori depend on the representation {{tmath|1= \rho }}), and a ''congruence subgroup'' of <math>\Gamma</math> 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 {{tmath|1= \pi_n(\Gamma) }}, and the level is defined.

=== Examples ===

The principal congruence subgroups of <math>\mathrm{SL}_d(\Z )</math> are the subgroups <math>\Gamma(n)</math> given by:
: <math>\Gamma(n) = \left\{(a_{ij}) \in \mathrm{SL}_d(\Z ): \forall i \, a_{ii} \equiv 1 \pmod n, \, \forall i \neq j \, a_{ij} \equiv 0 \pmod n \right\} </math>
the congruence subgroups then correspond to the subgroups of <math>\mathrm{SL}_d(\Z/n\Z )</math>.

Another example of arithmetic group is given by the groups <math>\mathrm{SL}_2(O)</math> where <math>O</math> is the [[ring of integers]] in a [[number field]], for example {{tmath|1= O = \Z[\sqrt 2] }}. Then if <math>\mathfrak p</math> is a [[prime ideal]] dividing a rational prime <math>p</math> the subgroups <math>\Gamma(\mathfrak p)</math> that is the kernel of the reduction map mod <math>\mathfrak p</math> is a congruence subgroup since it contains the principal congruence subgroup defined by reduction modulo {{tmath|1= p }}.

Yet another arithmetic group is the [[Siegel modular group]]s {{tmath|1= \mathrm{Sp}_{2g}(\Z) }}, defined by:
: <math>\mathrm{Sp}_{2g}(\Z) = \left\{ \gamma \in \mathrm{GL}_{2g}(\Z) : \ \gamma^{\mathrm{T}} \begin{pmatrix} 0 & I_g \\ -I_g & 0 \end{pmatrix} \gamma= \begin{pmatrix} 0 & I_g \\ -I_g & 0 \end{pmatrix} \right\}.</math>

Note that if <math>g = 1</math> then {{tmath|1= \mathrm{Sp}_2(\Z) = \mathrm{SL}_2(\Z) }}. The ''theta subgroup'' <math>\Gamma_{\vartheta}^{(n)}</math> of <math>\mathrm{Sp}_{2g}(\Z)</math> is the set of all <math>\left ( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right ) \in \mathrm{Sp}_{2g}(\Z)</math> such that both <math>AB^\top</math> and <math>CD^\top</math> have even diagonal entries.<ref>{{cite journal | last1 = Richter | first1 = Olav | year = 2000 | title = Theta functions of indefinite quadratic forms over real number fields | doi = 10.1090/s0002-9939-99-05619-1 | journal = [[Proceedings of the American Mathematical Society]] | volume = 128 | issue = 3| pages = 701–708 | doi-access = free}}</ref>

=== Property (&tau;) ===

The family of congruence subgroups in a given arithmetic group <math>\Gamma</math> always has property (&tau;) of Lubotzky&ndash;Zimmer.<ref>{{cite journal | last=Clozel | first=Laurent | title=Démonstration de la Conjecture &tau; | journal=Invent. Math. | volume=151 | date=2003 | issue=2 | pages=297&ndash;328 | language=French | doi=10.1007/s00222-002-0253-8| bibcode=2003InMat.151..297C | s2cid=124409226 }}</ref> This can be taken to mean that the [[Cheeger constant]] of the family of their [[Schreier coset graph]]s (with respect to a fixed generating set for {{tmath|1= \Gamma }}) is uniformly bounded away from zero, in other words they are a family of [[expander graph]]s. There is also a representation-theoretical interpretation: if <math>\Gamma</math> is a [[Lattice (discrete subgroup)|lattice]] in a [[Lie group]] {{tmath|1= G }} then property (&tau;) is equivalent to the non-trivial [[unitary representation]]s of {{tmath|1= G }} occurring in the spaces <math>L^2(G/\Gamma)</math> being bounded away from the trivial representation (in the [[Fell topology]] on the unitary dual of {{tmath|1= G }}). Property (&tau;) is a weakening of [[Kazhdan's property (T)]] which implies that the family of all finite-index subgroups has property (&tau;).

=== In ''S''-arithmetic groups ===

If <math>\mathbf G</math> is a <math>\Q </math>-group and <math>S = \{p_1,\ldots, p_r\}</math> is a finite set of primes, an <math>S</math>-arithmetic subgroup of <math>\mathbf G(\Q )</math> is defined as an arithmetic subgroup but using <math>\Z[1/p_1,\ldots, 1/p_r])</math> instead of {{tmath|1= \Z }}. The fundamental example is {{tmath|1= \operatorname{SL}_d(\Z [1/p_1,\ldots, 1/p_r]) }}.

Let <math>\Gamma_S</math> be an <math>S</math>-arithmetic group in an algebraic group {{tmath|1= \mathbf G \subset \operatorname{GL}_d }}. If <math>n</math> is an integer not divisible by any prime in {{tmath|1= S }}, then all primes <math>p_i</math> are invertible modulo <math>n</math> and it follows that there is a morphism {{tmath|1= \pi_n: \Gamma_S \to \mathrm{GL}_d(\Z/n\Z) }}. Thus it is possible to define congruence subgroups in {{tmath|1= \Gamma_S }}, whose level is always coprime to all primes in {{tmath|1= S }}.