In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a congruence subgroup of a
matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fa ...
with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
entries is a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
defined by congruence conditions on the entries. A very simple example would be
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
2 × 2 integer matrices of
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
1, in which the off-diagonal entries are ''even''. More generally, the notion of congruence subgroup can be defined for
arithmetic subgroup
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theor ...
s of
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
s; 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 {{unsourced, date=September 2022
In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a homomorphism ''h'' from ''G'' to a fini ...
. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
are essentially congruence subgroups.
Congruence subgroups of 2×2 matrices are fundamental objects in the classical theory of
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s; the modern theory of
automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s 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
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
.
Principal congruence subgroups
If
is an integer there is a homomorphism
induced by the reduction modulo
morphism
. The ''principal congruence subgroup of level
'' in
is the kernel of
, and it is usually denoted
. Explicitly it is described as follows:
:
This definition immediately implies that
is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
in
. The
strong approximation theorem (in this case an easy consequence of the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
) implies that
is surjective, so that the quotient
is isomorphic to
Computing the order of this finite group yields the following formula for the index:
:
where the product is taken over all prime numbers dividing
.
If
then the restriction of
to any finite subgroup of
is injective. This implies the following result:
:''If
then the principal congruence subgroups
are torsion-free. ''
The group
contains
and is not torsion-free. On the other hand, its image in
is torsion-free, and the quotient of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
by this subgroup is a sphere with three cusps.
Definition of a congruence subgroup
If
is a subgroup in
then it is called a ''congruence subgroup'' if there exists
such that it contains the principal congruence subgroup
. The ''level''
of
is then the smallest such
.
From this definition it follows that:
*Congruence subgroups are of finite index in
;
*The congruence subgroups of level
are in one-to-one correspondence with the subgroups of
Examples
The subgroups
, sometimes called the ''Hecke congruence subgroup'' of level
, is defined as the preimage by
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
. If
is prime then
is in natural bijection with the
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
over the finite field
, and explicit representatives for the (left or right) cosets of
in
are the following matrices:
:
The subgroups
are never torsion-free as they always contain the matrix
. There are infinitely many
such that the image of
in
also contains torsion elements.
The subgroups
are the preimage of the subgroup of unipotent matrices:
:
Their indices are given by the formula:
:
The ''theta subgroup''
is the congruence subgroup of
defined as the preimage of the cyclic group of order two generated by
. It is of index 3 and is explicitly described by:
:
These subgroups satisfy the following inclusions :
, as well as
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
is a nonconstant eigenfunction of the
Laplace-Beltrami operator on a congruence cover of the modular surface with eigenvalue
then
There is also a collection of distinguished operators called
Hecke operator In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic repr ...
s 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 form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s. 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
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of
in
has been investigated; one result from the 1970s, due to
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
,
Andrew Ogg and
John G. Thompson
John Griggs Thompson (born October 13, 1932) is an American mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992, and the Abel Prize in 2008.
...
is that the corresponding
modular curve
In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular grou ...
(the
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
resulting from taking the quotient of the hyperbolic plane by
) has
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
zero (i.e., the modular curve is a Riemann sphere)
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
''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
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
246320597611213317192329314147 ...
, he noticed that these were precisely the
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s of the size of ''M'', he wrote up a paper offering a bottle of
Jack Daniel's
Jack Daniel's is a brand of Tennessee whiskey. It is produced in Lynchburg, Tennessee, by the Jack Daniel Distillery, which has been owned by the Brown–Forman Corporation since 1956.
Packaged in square bottles, Jack Daniel's "Black Label" T ...
whiskey to anyone who could explain this fact – this was a starting point for the theory of
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979.
...
, 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
. In general, to give a definition one needs a
semisimple algebraic group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
defined over
and a faithful representation
, also defined over
from
into
; then an arithmetic group in
is any group
which is of finite index in the stabiliser of a finite-index sub-lattice in
.
Congruence subgroups
Let
be an arithmetic group: for simplicity it is better to suppose that
. As in the case of
there are reduction morphisms
. We can define a principal congruence subgroup of
to be the kernel of
(which may a priori depend on the representation
), and a ''congruence subgroup'' of
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
, and the level is defined.
Examples
The principal congruence subgroups of
are the subgroups
given by:
:
the congruence subgroups then correspond to the subgroups of
.
Another example of arithmetic group is given by the groups
where
is the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
in a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
, for example