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 formula_1. Principal congruence subgroups. If formula_2 is an  integer there is a homomorphism formula_3 induced by the reduction modulo formula_4 morphism formula_5. The "principal congruence subgroup of level formula_4" in formula_7 is the kernel of formula_8, and it is usually denoted formula_9. Explicitly it is described as follows: This definition immediately implies that formula_9 is a normal subgroup of finite index in formula_12. The strong approximation theorem (in this case an easy consequence of the Chinese remainder theorem) implies that formula_8 is surjective, so that the quotient formula_14 is isomorphic to formula_15 Computing the order of this finite group yields the following formula for the index: where the  product is taken over all prime numbers dividing formula_4. If formula_18 then the restriction of formula_8 to any finite subgroup of formula_12 is injective. This implies the following result: The group formula_23 contains formula_24 and is not torsion-free. On the other hand, its image in formula_25 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 formula_26 is a subgroup in formula_7 then it is called a "congruence subgroup" if there exists formula_2 such that it contains the principal congruence subgroup formula_9. The "level  formula_30 of formula_26 is then the smallest such formula_4. From this definition it follows that: Examples. The subgroups formula_36, sometimes called the Hecke congruence subgroup" of level formula_4, is defined as the preimage by formula_8 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 formula_4. If formula_42 is prime then formula_43 is in natural bijection with the projective line over the finite field formula_44, and explicit representatives for the (left or right) cosets of formula_45 in formula_12 are the following matrices: The subgroups  formula_36 are never torsion-free as they always contain the matrix formula_49. There are infinitely many formula_50 such that the image of formula_36 in formula_52 also contains torsion elements. The subgroups formula_53 are the preimage of the subgroup of unipotent matrices: Their indices are given by the formula: The "theta subgroup" formula_56 is the congruence subgroup of formula_12 defined as the preimage of the cyclic group of order two generated by formula_58. It is of index 3 and is explicitly described by: These subgroups satisfy the following inclusions : formula_60, as well as formula_61 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 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 formula_65 of formula_45 in formula_67 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 formula_65) 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 formula_69. In general, to give a definition one needs a semisimple algebraic group formula_70 defined over formula_71 and a faithful representation formula_72, also defined over formula_73 from formula_70 into formula_75 ; then an arithmetic group in formula_76 is any group formula_77 which is of finite index in the stabiliser of a finite-index sub-lattice in formula_78. Congruence subgroups. Let formula_12 be an arithmetic group: for simplicity it is better to suppose that formula_80. As in the case of formula_1 there are reduction morphisms formula_82. We can define a principal congruence subgroup of formula_12 to be the kernel of formula_8 (which may a priori depend on the representation formula_72), and a "congruence subgroup" of formula_12 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 formula_87, and the level is defined. Examples. The principal congruence subgroups of formula_88 are the subgroups formula_9 given by: the congruence subgroups then correspond to the subgroups of formula_91. Another example of arithmetic group is given by the groups formula_92 where formula_93 is the ring of integers in a number field, for example formula_94. Then if formula_95 is a prime ideal dividing a rational prime formula_42 the subgroups formula_97 which is the kernel of the reduction map mod formula_95 is a congruence  subgroup since it contains the principal congruence subgroup defined by reduction modulo formula_42. Yet another arithmetic group is the Siegel modular groups formula_100 defined by: Note that if formula_102 then formula_103 The "theta subgroup" formula_104 of formula_105 is the set of all formula_106 such that both formula_107 and formula_108 have even diagonal entries. Property (τ). The family of congruence subgroups in a given arithmetic group formula_12 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 formula_12 ) is uniformly bounded away from zero, in other words they are a family of expander graphs. There is also a representation-theoretical interpretation: if formula_12 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 formula_112 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 formula_70 is a formula_114-group  and formula_115 is a finite set of primes, an formula_116-arithmetic subgroup of formula_117 is defined as an arithmetic subgroup but using formula_118 instead of formula_119 The fundamental example is formula_120. Let formula_121 be an formula_116-arithmetic group in an algebraic group formula_123. If formula_4 is an integer not divisible by any prime in formula_116, then all primes formula_126 are invertible modulo formula_4 and it follows that there is a morphism formula_128 Thus it is possible to define congruence subgroups in formula_121, whose level is always coprime to all primes in formula_116. The congruence subgroup problem. Finite-index subgroups in  SL(Z). Congruence subgroups in formula_7 are finite-index subgroups: it is natural to ask whether they account for all finite-index subgroups in formula_12. 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 congruence kernel. 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 formula_1, when formula_18 all finite-index subgroups in formula_155 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 formula_156 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 formula_12. There is a topology on formula_12 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 formula_159 of formula_12, while the "congruence" topology gives rise to another completion formula_161. Both are profinite groups and there is a natural surjective morphism formula_162 (intuitively, there are fewer conditions for a Cauchy sequence to comply with in the congruence topology than in the profinite topology). The "congruence kernel" formula_163 is the kernel of this morphism, and the congruence subgroup problem  stated above amounts to whether formula_163 is trivial. The weakening of the conclusion then leads to the following problem. When the problem has a positive solution one says that formula_12 has the "congruence subgroup property". A conjecture generally attributed to Serre states that an irreducible arithmetic lattice in a semisimple Lie group formula_167 has the congruence subgroup property if and only if the real rank of formula_167 is at least 2; for example, lattices in formula_169 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 formula_170, the unitary groups formula_171 and the groups formula_172 (the isometry groups of a sesquilinear form over the Hamilton quaternions), plus the exceptional group formula_173 (see List of simple Lie groups). The current status of the congruence subgroup problem is as follows: 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 p-adic factors in the case of S-arithmetic groups) is at least 2: The case of inner forms of type formula_181 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 formula_169 or formula_186 with compact quotient. Congruence groups and adèle groups. The ring of adeles formula_187 is the restricted  product of all completions of formula_73 i.e. where the product is over the set formula_190 of all primes, formula_191 is the field of p-adic numbers and an element formula_192 belongs to the restricted product if and only if for almost all primes formula_42, formula_194 belongs to the subring formula_195 of p-adic integers. Given any algebraic group formula_70 over formula_114 the adelic algebraic group formula_198 is well-defined. It can be endowed with a canonical topology, which in the case where formula_70 is a linear algebraic group is the topology as a subset of formula_200. The finite adèles formula_201are  the restricted product of all non-archimedean completions (all p-adic fields). If formula_202 is an arithmetic group then its congruence subgroups are characterised by the following property: formula_203 is a congruence subgroup if and only if its closure formula_204 is a compact-open subgroup (compactness is automatic) and formula_205. In general the group formula_206 is equal to the congruence closure of formula_26 in formula_208 and the congruence topology on formula_12 is the induced topology as a subgroup of formula_210, in particular the congruence completion formula_161 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 formula_77 to be a congruence subgroup without explicit reference to a fixed arithmetic subgroup, by asking that it be equal to its congruence closure formula_213 Thus it becomes possible to study all congruence subgroups at once by looking at the discrete subgroup formula_214 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. Congruence subgroup