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Matrix group
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
Congruence_subgroup
Orientation-preserving mapping class group of the torus
video at right. Important subgroups of the modular group Γ, called congruence subgroups, are given by imposing congruence relations on the associated
Modular_group
Topics referred to by the same term
Ramanujan's congruences, congruences for the partition function, p(n), first discovered by Ramanujan in 1919 Congruence subgroup, a subgroup defined by
Congruence
Equivalence relation in algebra
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector
Congruence_relation
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
( N ) {\displaystyle \Gamma (N)} principal congruence subgroup of level N {\displaystyle N} . A subgroup Γ ⊆ S L 2 ( Z ) {\displaystyle \Gamma \subseteq
Maass_wave_form
Type of group in group theory
integer. These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture
Arithmetic_group
Series related to Ramanujan's pi formulas
Cooper found a general approach that used the underlying modular congruence subgroup Γ 0 ( n ) {\displaystyle \Gamma _{0}(n)} , while G. Almkvist has
Ramanujan–Sato_series
Discrete subgroup in a locally compact topological group
Conjecturally, arithmetic lattices in higher-rank groups have the congruence subgroup property but there are many lattices in S O ( n , 1 ) , S U ( n
Lattice_(discrete_subgroup)
Analytic function on the upper half-plane with a certain behavior under the modular group
poles of f in the closure of the fundamental region RΓ. For the congruence subgroup Γ0(N), the field of modular functions on X0(N) is generated by the
Modular_form
)} ; they belong to a more general class of finite-index subgroups, congruence subgroups. Any order in a quaternion algebra over Q {\displaystyle \mathbb
Arithmetic_Fuchsian_group
Mathematical group
group of degree n {\displaystyle n} . Its congruence subgroups, such as the principal congruence subgroup Γ n ( m ) = { γ ∈ Sp ( 2 n , Z ) : γ ≡ I
Symplectic_group
Type of generalization of periodic functions in Euclidean space
SL(2, R) or PSL(2, R), with the discrete subgroup being the modular group or one of its congruence subgroups; in this sense the theory of automorphic
Automorphic_form
Mathematician
Swinnerton-Dyer: if a modular form f(τ) is not modular for some congruence subgroup of the modular group, then the Fourier coefficients of f(τ) have
Yunqing_Tang
Algebraic object
the ring of modular forms M(Γ) is finitely generated when Γ is a congruence subgroup of SL(2, Z). In 2003, Lev Borisov and Paul Gunnells showed that the
Ring_of_modular_forms
Third letter of the Greek alphabet
machine The Feferman–Schütte ordinal Γ 0 {\displaystyle \Gamma _{0}} Congruence subgroups of the modular group of other arithmetic groups One of the Greeks
Gamma
Algebraic variety
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(2, Z). The term
Modular_curve
Mathematical function
{N}{d}}r_{d}\equiv 0{\pmod {24}},} then ηg is a weight k modular form for the congruence subgroup Γ0(N) (up to holomorphicity) where k = 1 2 ∑ 0 < d ∣ N r d . {\displaystyle
Dedekind_eta_function
Type of topological group
group is a discrete subgroup such that the Haar measure of the quotient space is finite. crystallographic point group congruence subgroup arithmetic group
Discrete_group
Number, approximately 1.618
})} is invariant under Γ ( 5 ) {\displaystyle \Gamma (5)} , a congruence subgroup of the modular group. Also for positive real numbers a {\displaystyle
Golden_ratio
Monster and modular connection
quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of the Hecke congruence subgroup Γ0(p) in SL(2,R). They found that
Monstrous_moonshine
finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem (see Lubotzky & Segal
Boundedly_generated_group
operators 〈d〉 are operators acting on the space of modular forms for the congruence subgroup Γ1(N), given by the action of a matrix (a b c δ) in Γ0(N) where δ
Diamond_operator
Concept in mathematics
7) triangle group, after quotienting by the center. The principal congruence subgroup defined by an ideal I ⊂ Z [ η ] {\displaystyle I\subset \mathbb {Z}
Hurwitz_quaternion_order
Australian-American mathematician
Swinnerton-Dyer: if a modular form f(τ) is not modular for some congruence subgroup of the modular group, then the Fourier coefficients of f(τ) have
Frank_Calegari
Subgroup invariant under conjugation
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation
Normal_subgroup
Group of mathematical theorems
generalise this to universal algebra, normal subgroups need to be replaced by congruence relations. A congruence on an algebra A {\displaystyle A} is an equivalence
Isomorphism_theorems
Mathematical term in group theory
of G is finite. The group G {\displaystyle G} has the congruence subgroup property: a subgroup H has finite index in G {\displaystyle G} if and only if
Grigorchuk_group
Algebraic structure
S} . Like any equivalence relation, a semigroup congruence ∼ {\displaystyle \sim } induces congruence classes [ a ] = { x ∈ S ∣ x ∼ a } {\displaystyle
Semigroup
Israeli mathematician and former politician
Study in Princeton a year long program on "Pro-finite groups and the congruence subgroup problem". In 2006, he got an honorary degree from the University
Alexander_Lubotzky
Mathematical concept
arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties
Shimura_variety
Theorems that help decompose a finite group based on prime factors of its order
groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a normal subgroup. Burnside's pa qb theorem
Sylow_theorems
In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let G {\displaystyle
Subgroup_growth
theory, a subgroup of a group is said to have the Congruence Extension Property or to be a CEP subgroup if every congruence on the subgroup lifts to a
CEP_subgroup
Compact Riemann surface of genus 3
action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the ideal I = ⟨ η − 2 ⟩ {\displaystyle I=\langle
Klein_quartic
Indian-American mathematician (born 1935)
including the study of lattices in semi-simple Lie groups and the congruence subgroup problem. In 1976, Prasad received his Ph.D. from the University of
Gopal_Prasad
Sum of inverse squares of natural numbers
2 ( Z p , p ) {\displaystyle SL_{2}(\mathbb {Z} _{p},p)} is the congruence subgroup modulo p {\displaystyle p} . Since each of the coordinates x , y
Basel_problem
Bulgarian mathematician (born 1986)
form f ( τ ) {\displaystyle f(\tau )} is not modular for some congruence subgroup of the modular group, then the Fourier coefficients of f ( τ )
Vesselin_Dimitrov
Group of units of the ring of integers modulo n
n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
Construction in group theory
elements mod n; the kernels are called the principal congruence subgroups. A noteworthy subgroup of the projective general linear group PGL(2, Z) (and
Projective_linear_group
Non-singular cubic surface in mathematics
principal congruence subgroup of the Hilbert modular group of the field Q(√5). The quotient of the Hilbert modular group by its level 2 congruence subgroup is
Clebsch_surface
conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1 / 4 {\displaystyle 1/4} . Selberg's orthogonality
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
group Complete group Complex reflection group Congruence subgroup Continuous symmetry Frattini subgroup Growth rate Heisenberg group, discrete Heisenberg
List_of_group_theory_topics
(Szeged) 24 (1963), 34–59. Pálfy and Pudlák. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis
Finite lattice representation problem
Finite_lattice_representation_problem
Mathematical conjecture about eigenvalues
that the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1/4. Selberg showed that the eigenvalues are at least 3/16
Selberg's_1/4_conjecture
Lattice whose elements are the subgroups of a given group
example), the lattice of congruences is modular (Kearnes & Kiss 2013). Lattice-theoretic information about the lattice of subgroups can sometimes be used
Lattice_of_subgroups
Concept in mathematics
The kernel of Φ n {\displaystyle \Phi _{n}} is usually called a congruence subgroup of Mod ( S ) {\displaystyle \operatorname {Mod} (S)} . It is a
Mapping class group of a surface
Mapping_class_group_of_a_surface
Mathematical theorem
acting on a vector space of cusp forms of a given weight, for a given congruence subgroup of the modular group. Here the trace of the identity operator is
Selberg_trace_formula
36 mathematical problems stated in 1955
Reihen durch ihre Funktionalgleichung", which involves not only congruence subgroups of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} but also
Taniyama's_problems
British mathematician (1927–2018)
conjectured that if a modular form f(τ) is not modular for some congruence subgroup of the modular group, then the Fourier coefficients of f(τ) have
Peter_Swinnerton-Dyer
Branch of mathematics that studies abstract algebraic structures
the modular group PSL2 (R) and a chosen congruence subgroup by a semisimple Lie group G and a discrete subgroup Γ. Just as modular forms can be viewed
Representation_theory
mathematics, an Igusa group or Igusa subgroup is a subgroup of the Siegel modular group defined by some congruence conditions. They were introduced by
Igusa_group
Bass, Milnor & Serre (1967), who used them in their solution of the congruence subgroup problem. Suppose that A is a Dedekind domain and q is a non-zero
Mennicke_symbol
Concept in math
space G/H. In number theory, the Hecke algebra corresponding to a congruence subgroup Γ of the modular group is spanned by elements of the double coset
Double_coset
Generalized manifold
the stabiliser of a vertex is a subgroup of order 21 and Ψ is injective on this subgroup. Thus if the congruence subgroup Γ0 is defined as the inverse image
Orbifold
Part of the theory of modular forms
cusp form 'new' at a given level N, where the levels are the nested congruence subgroups: Γ 0 ( N ) = { ( a b c d ) ∈ SL ( 2 , Z ) : c ≡ 0 ( mod N ) } {\displaystyle
Atkin–Lehner_theory
Mathematical group of the homotopy classes of loops in a topological space
K={\text{SO}}(2)} and Γ {\displaystyle \Gamma } any torsion-free congruence subgroup of the modular group SL ( 2 , Z ) {\displaystyle {\text{SL}}(2,\mathbb
Fundamental_group
Form of differential geometry
constants. Thus, Hurwitz surfaces Σg defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound s y s
Systolic_geometry
Hierarchical level in biological classification
Carolina A; Aylward, Frank O (26 August 2021). "Phylogenetic Signal, Congruence, and Uncertainty across Bacteria and Archaea". Molecular Biology and Evolution
Taxonomic_rank
Group obtained by aggregating similar elements of a larger group
class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group
Quotient_group
Important problem in lattice theory
In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other
Congruence_lattice_problem
Local zeta function Weil conjectures Modular form modular group Congruence subgroup Hecke operator Cusp form Eisenstein series Modular curve Ramanujan–Petersson
List_of_number_theory_topics
tensor categories are congruence subgroups of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} . Since congruence subgroups all have finite index
Schauenburg–Ng_theorem
splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding
Hurwitz_surface
Theory in supersymmetric gauge theory
(2)<\mathrm {SL} (2,\mathbb {Z} )} is the second principal congruence subgroup, the subgroup of matrices congruent to 1 mod 2, generated by M ∞ = ( − 1
Seiberg–Witten_theory
Smallest monoid that recognizes a formal language
_{S}xt\ } for all x ∈ M {\displaystyle x\in M} . The syntactic congruence or Myhill congruence is defined as s ≡ S t ⇔ ( ∀ x , y ∈ M : x s y ∈ S ⇔ x
Syntactic_monoid
In mathematics, the Shimura subgroup Σ(N) is a subgroup of the Jacobian of the modular curve X0(N) of level N, given by the kernel of the natural map
Shimura_subgroup
Mathematical concept
construction of the Moduli stack of elliptic curves. This is related to the congruence subgroup Γ ( 1 ) {\displaystyle \Gamma (1)} in the following way: M ( [ Γ
J-line
Connects non-singular algebraic curves with compact Riemann surfaces
half-plane and Γ is a subgroup of finite index in the modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is not the
Belyi's_theorem
Unsolved problem in mathematics
general Ramanujan–Petersson conjecture for holomorphic cusp forms for congruence subgroups was proposed in Petersson (1930) and has a similar formulation: a
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Modular function in mathematics
2307/2371313. JSTOR 2371313. MR 1507331. Cummins, Chris J. (2004). "Congruence subgroups of groups commensurable with PSL(2,Z)$ of genus 0 and 1". Experimental
J-invariant
Type of group in mathematics
connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of
Orthogonal_group
Ring without nonzero zero divisors
the case where K is the ring of p-adic integers and G is the pth congruence subgroup of GL(n, Z). Zero divisors have a topological interpretation, at
Domain_(ring_theory)
constructed as the principal congruence subgroup of the (2,3,7) triangle group in a suitable tower of principal congruence subgroups. Here the choices of quaternion
MacBeath_surface
Theorem in group theory
More generally an analogous result that concerns congruence relations instead of normal subgroups holds for any algebraic structure. Modular lattice
Correspondence_theorem
Mathematics term
representation, although SL(2,Z) has property (τ) with respect to principal congruence subgroups, by Selberg's theorem. Noncompact solvable groups. Nontrivial free
Kazhdan's_property_(T)
Disjoint, equal-size subsets of a group's underlying set
mZ + m = m(Z + 1) = mZ. The coset (mZ + a, +) is the congruence class of a modulo m. The subgroup mZ is normal in Z, and so, can be used to form the quotient
Coset
Result of partitioning the elements of an algebraic structure using a congruence relation
algebraic structure using a congruence relation. Quotient algebras are also called factor algebras. Here, the congruence relation must be an equivalence
Quotient_(universal_algebra)
In mathematics, a Riemann surface
(2016-10-01). "Bolza Quaternion Order and Asymptotics of Systoles Along Congruence Subgroups". Experimental Mathematics. 25 (4): 399–415. arXiv:1405.5454. doi:10
Bolza_surface
South Korean American mathematician
exponential mixing and resonance free regions for convex cocompact congruence subgroups of SL_2(Z), Journal of the American Mathematical Society, vol. 29
Hee_Oh
Three Riemann surfaces with same symmetry
splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding
First_Hurwitz_triplet
Topics referred to by the same term
Γ0 or Gamma 0 may refer to: Feferman–Schütte ordinal Hecke congruence subgroup, Γ0(n) the multiple gamma function, Γn, for n = 0, as used in an inductive
Γ₀
Algebraic variety that is a moduli space for principally polarized abelian varieties
quotient of the Siegel upper half-space by the action of the principal congruence subgroup of level n of a symplectic group. A Siegel modular variety may also
Siegel_modular_variety
Algebraic surface in mathematics
Hilbert modular group may be replaced by some subgroup of finite index, such as a congruence subgroup. One can extend the Hilbert modular group by a
Hilbert_modular_variety
Elements taken to zero by a homomorphism
whether a homomorphism is injective. In these cases, the kernel is a congruence relation. Kernels allow defining quotient objects (also called quotient
Kernel_(algebra)
Integer that is a perfect square modulo some integer
{\displaystyle (\mathbb {Z} /p\mathbb {Z} )} . In other words, every congruence class except zero modulo p has a multiplicative inverse. This is not true
Quadratic_residue
Σ g {\displaystyle \Sigma _{g}} defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound s y s
Systoles_of_surfaces
number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer. 3. May denote a logical equivalence. ≅ 1. May denote
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Special functions of several complex variables
)=\operatorname {SL} (2,\mathbb {Z} )} . The n-dimensional analogue of the congruence subgroups is played by ker { Sp ( 2 n , Z ) → Sp ( 2 n , Z / k Z ) }
Theta_function
Mathematical function
Bruinier and Funke (for arbitrary weights, multiplier systems, and congruence subgroups) that this map is surjective. Consequently, there is an exact sequence
Harmonic_Maass_form
Index of articles associated with the same name
relation. Not every congruence on a semigroup is associated with an ideal, so a simple semigroup may have nontrivial congruences. A semigroup with no
Simple_(abstract_algebra)
Borel (1964, 1966). If C is the quotient of the upper half plane by a congruence subgroup of SL2(Z), then the Baily–Borel compactification of C is formed by
Baily–Borel_compactification
Group of transformations under which the object is invariant
is G = Sym(X). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers
Symmetry_group
Problem of inverting exponentiation in groups
multiplication modulo the prime p {\displaystyle p} . Its elements are non-zero congruence classes modulo p {\displaystyle p} , and the group product of two elements
Discrete_logarithm
x , r + ε {\displaystyle G(k)_{x,r+}=G(k)_{x,r+\varepsilon }} . Congruence subgroup Bruhat & Tits 1972, Section 6.4. Moy & Prasad 1994. Hakim & Murnaghan
Moy–Prasad_filtration
Chinese-American mathematician
Parsa; Rowell, Eric C.; Zhang, Qing; Wang, Zhenghan (2018-07-16). "Congruence Subgroups and Super-Modular Categories". Pacific Journal of Mathematics. 296
Zhenghan_Wang
Theorem on modular exponentiation
not prime. The converse of Euler's theorem is also true: if the above congruence is true, then a {\displaystyle a} and n {\displaystyle n} must be coprime
Euler's_theorem
Lattice (discrete subgroup) Frieze group Wallpaper group Space group Crystallographic group Fuchsian group Modular group Congruence subgroup Kleinian group
List_of_Lie_groups_topics
Hecke algebra Hecke algebra (disambiguation) Hecke character Hecke congruence subgroup Hecke correspondence Hecke eigenform Hecke group Hecke L-function
List of things named after Erich Hecke
List_of_things_named_after_Erich_Hecke
"Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups". J. Differential Geom. 76 (3): 399–422. arXiv:math.DG/0505007.
(2,3,7)_triangle_group
Subgroup of a group in mathematics
Conversely, any retract which is a normal subgroup is a direct factor. Every retract has the congruence extension property. Every regular factor, and
Retract_(group_theory)
CONGRUENCE SUBGROUP
CONGRUENCE SUBGROUP
Boy/Male
Bengali, Gujarati, Hindu, Indian, Sanskrit
Union; Noble; Confluence
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Confluence of Three Sacred Rivers
Boy/Male
Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Confluence of Three Sacred River Ganga, Yamuna and Saraswati
Boy/Male
Indian, Modern
The Confluence of Three Rivers; Great Hunter
Girl/Female
Tamil
Triveni | தà¯à®°à®¿à®µà¯‡à®£à¯€
Confluence of three sacred rivers
Triveni | தà¯à®°à®¿à®µà¯‡à®£à¯€
Boy/Male
Hindu
Confluence of Ganga Jamuna Saraswati
Boy/Male
Tamil
Confluence of Ganga Jamuna Saraswati
CONGRUENCE SUBGROUP
CONGRUENCE SUBGROUP
Boy/Male
Hindu
Boy/Male
Tamil
Dwaraka | தà¯à®µà®¾à®°à®•à®¾
Capital, Lord krishnas kingdom
Boy/Male
Indian, Punjabi, Sikh
Lord of the Team
Boy/Male
Hindu, Indian, Marathi
Known; Understood
Girl/Female
Indian
Gentle, Patient
Girl/Female
Muslim
Fourth.
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu
One who Bring Happiness; Joyful; Happy; Always Smiling
Girl/Female
German, Hungarian
Renowned Battle; Famous Battle
Girl/Female
Indian, Sikh
Soft; Pretty
Girl/Female
British, English, German
Willow; Untamed
CONGRUENCE SUBGROUP
CONGRUENCE SUBGROUP
CONGRUENCE SUBGROUP
CONGRUENCE SUBGROUP
CONGRUENCE SUBGROUP
n.
Reduction to congruence or consistency; removal of inconsistency; harmony.
n.
Want of congruence; incongruity.
n.
Suitableness of one thing to another; agreement; consistency.
n.
A moving, flowing, or running together; confluence.
n.
Any running together of separate streams or currents; the act of meeting and crowding in a place; hence, a crowd; a concourse; an assemblage.
n.
A concurence or general tendency, as of circumstances, to one event, as if by agreement.
prep.
Violent confluence.
n.
A subdivision of a group, as of animals.
n.
The act of flowing together; the meeting or junction of two or more streams; the place of meeting.
n.
Congruence.
a.
Possessing congruity; suitable; agreeing; corresponding.