Congruence subgroup
Encyclopedia
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 2x2 integer matrices of determinant
1, such that the off-diagonal entries are even.
An importance class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the special linear group
SL(n, Z) we can reduce the entries to modular arithmetic
in Z/NZ for any N >1, which gives a homomorphism
of groups. The kernel of this reduction map is an example of a congruence subgroup – the condition is that the diagonal entries are congruent to 1 mod N, and the off-diagonal entries be congruent to 0 mod N (divisible by N), and is known as a .
In the case n=2 we are talking then about a subgroup of the modular group (up to the quotient by {I,-I} taking us to the corresponding projective group): the kernel of reduction is called Γ(N) and plays a big role in the theory of modular form
s. Further, we may take the inverse image of any subgroup (not just {e}) and get a congruence subgroup: the subgroups Γ0(N) important in modular form theory are defined in this way, from the subgroup of mod N 2x2 matrices with 1 on the diagonal and 0 below it.
More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic group
s; that is, those for which we have a notion of 'integral structure' respected by the subgroup, and so some general idea of what 'congruence' means.
actually congruence subgroups? This is not in general true, and non-congruence subgroups exist. It is however an interesting question to understand when these examples are possible. This problem about the classical group
s was resolved by . .
It can be posed in topological terms: if Γ is some arithmetic group, there is a topology on Γ for which a base of neighbourhoods of {e} is the set of subgroups of finite index; and there is another topology defined in the same way using only congruence subgroups. We can ask whether those are the same topologies; equivalently, if they give rise to the same completions. The subgroups of finite index give rise to the completion of Γ as a pro-finite group
. If there are essentially fewer congruence subgroups, the corresponding completion of Γ can be bigger (intuitively, there are fewer conditions for a Cauchy sequence
to comply with). Therefore the problem can be posed as a relationship of two compact
topological group
s, with the question reduced to calculation of a possible kernel
. The solution by Hyman Bass
, Jean-Pierre Serre
and John Milnor
involved an aspect of algebraic number theory
linked to K-theory
.
The use of adele
methods for automorphic representations (for example in the Langlands program
) implicitly uses that kind of completion with respect to a congruence subgroup topology - for the reason that then all congruence subgroups can then be treated within a single group representation
. This approach - using a group G(A) and its single quotient G(A)/G(Q) rather than looking at many G/Γ as a whole system - is now normal in abstract treatments.
has proved basic in much research, in number theory
, and in other areas such as monstrous moonshine
.
r, the modular group Γ0(r) is defined as follows:
It can be shown that for a prime number
p, the set
(where Sτ = −1/τ and Tτ = τ + 1) is a fundamental region of Γ0(r).
The normalizer Γ0(p)+ of Γ0(p) in SL(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 Γ0(p)+) has genus
zero (the modular curve is an elliptic curve
) 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 factor
s 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.
of the modular group Γ. It can be characterized as the set of linear Möbius transformations w that satisfy
with a and d being odd and b and c being even. That is, it is the congruence subgroup that is the kernel of reduction modulo 2, otherwise known as Γ(2).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a congruence subgroup of a matrix group
Matrix group
In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring R...
with integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
entries is a subgroup
Subgroup
In group theory, 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 of G if the restriction of * to H x H is a group operation on H...
defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
1, such that the off-diagonal entries are even.
An importance class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
SL(n, Z) we can reduce the entries to modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
in Z/NZ for any N >1, which gives a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
- SL(n, Z) → SL(n, Z/N·Z)
of groups. The kernel of this reduction map is an example of a congruence subgroup – the condition is that the diagonal entries are congruent to 1 mod N, and the off-diagonal entries be congruent to 0 mod N (divisible by N), and is known as a .
In the case n=2 we are talking then about a subgroup of the modular group (up to the quotient by {I,-I} taking us to the corresponding projective group): the kernel of reduction is called Γ(N) and plays a big role in the theory of modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s. Further, we may take the inverse image of any subgroup (not just {e}) and get a congruence subgroup: the subgroups Γ0(N) important in modular form theory are defined in this way, from the subgroup of mod N 2x2 matrices with 1 on the diagonal and 0 below it.
More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s; that is, those for which we have a notion of 'integral structure' respected by the subgroup, and so some general idea of what 'congruence' means.
Congruence subgroups and topological groups
Are all subgroups of finite indexIndex of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
actually congruence subgroups? This is not in general true, and non-congruence subgroups exist. It is however an interesting question to understand when these examples are possible. This problem about the classical group
Classical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s was resolved by . .
It can be posed in topological terms: if Γ is some arithmetic group, there is a topology on Γ for which a base of neighbourhoods of {e} is the set of subgroups of finite index; and there is another topology defined in the same way using only congruence subgroups. We can ask whether those are the same topologies; equivalently, if they give rise to the same completions. The subgroups of finite index give rise to the completion of Γ as a pro-finite group
Pro-finite group
In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.- Definition :...
. If there are essentially fewer congruence subgroups, the corresponding completion of Γ can be bigger (intuitively, there are fewer conditions for a Cauchy sequence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
to comply with). Therefore the problem can be posed as a relationship of two compact
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
s, with the question reduced to calculation of a possible kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
. The solution by Hyman Bass
Hyman Bass
Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...
, Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
and John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
involved an aspect of algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
linked to K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
.
The use of adele
Adele ring
In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....
methods for automorphic representations (for example in the Langlands program
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
) implicitly uses that kind of completion with respect to a congruence subgroup topology - for the reason that then all congruence subgroups can then be treated within a single group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
. This approach - using a group G(A) and its single quotient G(A)/G(Q) rather than looking at many G/Γ as a whole system - is now normal in abstract treatments.
Congruence subgroups of the modular group
Detailed information about the congruence subgroups of the modular group ΓModular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
has proved basic in much research, in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, and in other areas such as monstrous moonshine
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
.
Modular group Γ0(r)
For a given positive integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
r, the modular group Γ0(r) is defined as follows:
It can be shown that for a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p, the set
(where Sτ = −1/τ and Tτ = τ + 1) is a fundamental region of Γ0(r).
The normalizer Γ0(p)+ of Γ0(p) in SL(2,R) has been investigated; one result from the 1970s, due to Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, Andrew Ogg
Andrew Ogg
Andrew Pollard Ogg is an American mathematician, a professor emeritus of mathematics at the University of California, Berkeley....
and John G. Thompson
John G. Thompson
John Griggs Thompson is a 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 2008 Abel Prize....
is that the corresponding modular curve
Modular curve
In number theory and algebraic geometry, a modular curve Y 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 group of integral 2×2 matrices SL...
(the Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
resulting from taking the quotient of the hyperbolic plane by Γ0(p)+) has genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
zero (the modular curve is an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
) if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
, he noticed that these were precisely the prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
s of the size of M, he wrote up a paper offering a bottle of Jack Daniel's
Jack Daniel's
Jack Daniel's is a brand of sour mash Tennessee whiskey that is among the world's best-selling liquors. It is known for its square bottles and black label. As of November, 2007, one blogger was claiming that it was the best-selling whiskey in the world. It is produced in Lynchburg, Tennessee by...
whiskey to anyone who could explain this fact – this was a starting point for the theory of Monstrous moonshine
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
, which explains deep connections between modular function theory and the monster group.
Modular group Λ
The modular group Λ (also called the theta subgroup) is another subgroupSubgroup
In group theory, 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 of G if the restriction of * to H x H is a group operation on H...
of the modular group Γ. It can be characterized as the set of linear Möbius transformations w that satisfy
with a and d being odd and b and c being even. That is, it is the congruence subgroup that is the kernel of reduction modulo 2, otherwise known as Γ(2).