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Orientation-preserving mapping class group of the torus
In mathematics, the modular group is the projective special linear group PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} of 2 × 2
Modular_group
In mathematics, a Picard modular group, studied by Picard (1881), is a group of the form SU(J,L), where L is a 3-dimensional lattice over the ring of
Picard_modular_group
Analytic function on the upper half-plane with a certain behavior under the modular group
of the modular group and a growth condition. A modular form is a special case of an automorphic form, which are functions defined on Lie groups that transform
Modular_form
Representation of a modular tensor category
In mathematics, the modular group representation (or simply modular representation) of a modular tensor category C {\displaystyle {\mathcal {C}}} is a
Modular_group_representation
Group whose operation is a composition of braids
{R} )} . Furthermore, the modular group has trivial center, and thus the modular group is isomorphic to the quotient group of B 3 {\displaystyle B_{3}}
Braid_group
Algebraic variety
subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ)
Modular_curve
Special modular forms
In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function
Hilbert_modular_form
Concept in mathematics
topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface
Mapping class group of a surface
Mapping_class_group_of_a_surface
Construction in group theory
group only being the orientation-preserving transformations. PGL and PSL can also be defined over a ring, with an important example being the modular
Projective_linear_group
Matrix group
setting in which congruence subgroups can be studied is that of the modular group S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )} . If
Congruence_subgroup
Algebraic surface in mathematics
of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking
Hilbert_modular_variety
Modular electric car platform
The Volkswagen Group MEB platform (German: Modularer E-Antriebs Baukasten, 'modular electric-drive toolkit') is a modular car platform for electric cars
Volkswagen_Group_MEB_platform
Complex-differentiable part of a Maass wave function
mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight
Mock_modular_form
Motor vehicle platform
The Volkswagen Group MQB platform is the company's strategy for shared modular design construction of its transverse, front-engine, front-wheel-drive
Volkswagen_Group_MQB_platform
Symmetric holomorphic function
of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over
Modular_lambda_function
Motor vehicle platform
The Volkswagen Group MLB platform is the company's platform strategy, announced in 2012, for shared modular construction of its longitudinal, front-engined
Volkswagen_Group_MLB_platform
Mathematical group
{\displaystyle n=1} , this group coincides with the modular group SL ( 2 , Z ) {\displaystyle \operatorname {SL} (2,\mathbb {Z} )} . The group Sp ( 2 n , Z )
Symplectic_group
Space of complex matrices with positive definite imaginary part
axis of the Siegel half-plane Paramodular group, a generalization of the Siegel modular group Siegel modular form, a type of automorphic form defined on
Siegel_upper_half-space
Series representing modular forms
are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series
Eisenstein_series
Finite group
to this group as the "modular group of order 16", as its lattice of subgroups is modular. In this article this group will be called the modular maximal-cyclic
Quasidihedral_group
\backslash \mathbb {H} ^{2}} , where Γ {\displaystyle \Gamma } is the modular group, the Selberg zeta-function is of special interest. For this special
Selberg_zeta_function
Group of 𝑛 × 𝑛 invertible matrices
well as the study of polynomials. The modular group may be realised as a quotient of the special linear group SL ( 2 , Z ) {\displaystyle \operatorname
General_linear_group
Integer side lengths of a right triangle
This set forms a group, since the inverse of a matrix in Γ is again in Γ, as is the product of two matrices in Γ. The modular group acts on the collection
Pythagorean_triple
Group of real 2×2 matrices with unit determinant
modular group PSL(2, Z). Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group)
SL2(R)
Computation modulo a fixed integer
In mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when
Modular_arithmetic
name comes from the classical name modular group of this group, as in modular form theory. In string theory, modular invariance is an additional requirement
Modular_invariance
Periodic set of points
automorphism group of the lattice L ≅ Z 2 {\displaystyle L\cong \mathbb {Z} ^{2}} , which is a double cover of the well-studied modular group S L 2 ( Z )
Lattice_(group)
Type of monoidal category
A modular tensor category (or modular fusion category) is a type of monoidal category that plays a role in the areas of topological quantum field theory
Modular_tensor_category
Group of unitary complex matrices with determinant of 1
{\displaystyle \mathbb {C} } . An important example of this type of group is the Picard modular group SU ( 2 , 1 ; Z [ i ] ) {\displaystyle \operatorname {SU}
Special_unitary_group
Modular function in mathematics
mathematics, the j-invariant or j function is a modular function of weight zero for the special linear group SL ( 2 , Z ) {\displaystyle \operatorname {SL}
J-invariant
Type of generalization of periodic functions in Euclidean space
topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms
Automorphic_form
Parametrizes complex structures on a surface
compactification is acted on continuously by the modular group. In particular any element of the modular group has a fixed point in Thurston's compactification
Teichmüller_space
a paramodular group is a special sort of arithmetic subgroup of the symplectic group. It is a generalization of the Siegel modular group, and has the same
Paramodular_group
Discrete subgroup of the real projective special linear group of dimension 2
any disk in the Riemann sphere. Indeed, even the modular group PSL(2,Z), which is a Fuchsian group, does not act discontinuously on the real number line;
Fuchsian_group
Type of Kac–Moody algebras
characters of their representations transform amongst themselves under the modular group. If g {\displaystyle {\mathfrak {g}}} is a finite-dimensional simple
Affine_Lie_algebra
Mathematical function
mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of
Dedekind_eta_function
Operation that combines groups
the action of the modular group on a certain tessellation of the hyperbolic plane, it follows from this theory that the modular group is isomorphic to
Free_product
Topological space
for the action of the modular group Γ on the upper half-plane H. This famous diagram appears in all classical books on modular functions. (It was probably
Fundamental_domain
indicates a long-wheelbase variant. In 2007, Volkswagen Group introduced a more flexible "modular component system" architecture on which to base future
List of Volkswagen Group platforms
List_of_Volkswagen_Group_platforms
Chinese mathematician
Atkin and Swinnerton-Dyer: if a modular form f(τ) is not modular for some congruence subgroup of the modular group, then the Fourier coefficients of
Yunqing_Tang
Unsolved problem in mathematics
sublattices permuted by the modular group PSL(2, Z), which is based on changes of basis for Λ. Let j denote the elliptic modular function of Felix Klein.
Inverse_Galois_problem
Picard modular surface, studied by Picard (1881), is a complex surface constructed as a quotient of the unit ball in C2 by a Picard modular group. Picard
Picard_modular_surface
Type of group in group theory
_{2}(\mathbb {Z} )} , is called the modular group as it is related to the modular curve. Similar examples are the Siegel modular groups S p 2 g ( Z ) {\displaystyle
Arithmetic_group
Branch of mathematics that studies the properties of groups
Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained
Group_theory
mathematics, the Schauenbug–Ng theorem is a theorem about the modular group representations of modular tensor categories proved by Siu-Hung Ng and Peter Schauenburg
Schauenburg–Ng_theorem
Concept in information theory
an automorphic function with respect to a particular subgroup of the modular group. In theoretical computer science, the min-entropy is used in the context
Rényi_entropy
Linear operator acting on modular forms
the space of modular forms of a given weight. Another way to express Hecke operators is by means of double cosets in the modular group. In the contemporary
Hecke_operator
Function with unusual fractal properties
integer coefficients, the monoid may be regarded as a subset of the modular group PSL(2, Z). The question mark function provides a one-to-one mapping
Minkowski's question-mark function
Minkowski's_question-mark_function
26-dimensional string theory
to the moduli space of the torus, is any fundamental domain for the modular group P S L ( 2 , Z ) {\displaystyle PSL(2,\mathbb {Z} )} acting on the upper
Bosonic_string_theory
subgroup is a modular subgroup, that is, a modular element in the lattice of subgroups. This follows from the modular property of groups. If all subgroups
Quasinormal_subgroup
Plane algebraic curve
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0, such that (x, y) = (j(nτ), j(τ))
Classical_modular_curve
Type of topological group
plane to the whole space. The modular group PSL(2,Z) is thought of as a discrete subgroup of PSL(2,R). The modular group is a lattice in PSL(2,R), but
Discrete_group
Electronic musical instrument
The Moog synthesizer (/moʊɡ/ MOHG) is a modular synthesizer invented by the American engineer Robert Moog in 1964. Moog's company, R. A. Moog Co., produced
Moog_synthesizer
Mathematical concept
another proof of their hyperbolicity. An interesting example is the modular group G = S L 2 ( Z ) {\displaystyle G=\mathrm {SL} _{2}(\mathbb {Z} )} :
Hyperbolic_group
mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group G is called an
Iwasawa_group
Rational function of the form (az + b)/(cz + d)
important discrete subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian
Möbius_transformation
an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form that is an eigenvector for all Hecke operators Tm, m = 1
Eigenform
Type of mathematical generalization
the symmetries of Fuchsian groups in general (see, for example Indra's pearls and the Apollonian gasket) and the modular group in particular. The connection
Q-analog
Sub-field of mathematics
In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic (usually dividing
Modular_invariant_theory
Motor vehicle platform
Volkswagen Group MSB platform (Modularer Standardantriebsbaukasten, modular standard drivetrain matrix) is the company's strategy for shared modular design
Volkswagen_Group_MSB_platform
Group realized geometrically by reflections across the sides of a triangle
rotational triangle group (2,q,∞), and maps onto all triangle groups (2,q,n) by adding the relation Tn = 1 the modular group is the Hecke group H3. In Grothendieck's
Triangle_group
Mathematical group
exhibit the (2,3,7) triangle group as a specific Fuchsian group in SL(2,R), specifically as a quotient of the modular group. This can be visualized by the
(2,3,7)_triangle_group
Set with associative invertible operation
sporadic group is called the monster group. The monstrous moonshine conjectures, proved by Richard Borcherds, relate the monster group to certain modular functions
Group_(mathematics)
Type of nuclear fission reactor
A small modular reactor (SMR) is an emergent class of nuclear fission reactors with a rated electrical power of less than 300 megawatts (MWe), which use
Small_modular_reactor
Pakistani mathematician
"he has laid the foundation of coset diagrams for the modular group", to study the actions of groups on various spaces and projective lines over Galois fields
Qaiser_Mushtaq
Algebraic variety that is a moduli space for principally polarized abelian varieties
action of a symplectic group. Complex analytic spaces have naturally associated algebraic varieties by Serre's GAGA. The Siegel modular variety Ag(n), which
Siegel_modular_variety
Studies linear representations of finite groups over fields of positive characteristic
finite groups over a field K of positive characteristic p, necessarily a prime number. As well as having applications to group theory, modular representations
Modular_representation_theory
sets (in the upper halfplane), and is a modular form of weight 2k for Γ. Note that, when Γ is the full modular group and n = 0, one obtains the Eisenstein
Poincaré series (modular form)
Poincaré_series_(modular_form)
Monster and modular connection
moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular the j function. The initial numerical observation
Monstrous_moonshine
subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article 'factor of automorphy'
Automorphic_factor
Type of mathematical group
arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group P S L 2 ( Z ) {\displaystyle \mathrm {PSL} _{2}(\mathbb {Z} )}
Arithmetic_Fuchsian_group
Group of matrices with determinant 1
}} . SL(2, R) SL(2, C) Modular group (PSL(2, Z)) Projective linear group Conformal map Representations of classical Lie groups Hall 2015 Section 2.5 Hall
Special_linear_group
Third letter of the Greek alphabet
{\displaystyle \Gamma _{0}} Congruence subgroups of the modular group of other arithmetic groups One of the Greeks in mathematical finance In Medieval music
Gamma
Operation in group theory
modular sublattice. A group in which every subgroup permutes is called an Iwasawa group. The subgroup lattice of an Iwasawa group is thus a modular lattice
Product_of_group_subsets
Connects non-singular algebraic curves with compact Riemann surfaces
modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion that any such curve is a modular
Belyi's_theorem
French mathematician (1856–1941)
Recherche Scientifique. OCLC 4615520. Émile Picard Medal Picard modular group Picard modular surface Picard horn Bregille Funicular Hadamard, J. (1942). "Emile
Émile_Picard
Type of lattice in mathematical order theory
mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, Modular law a ≤ b implies a ∨ (x ∧ b) =
Modular_lattice
American singer (born 1985)
singers, including members of Au Revoir Simone and Class Actress. The modular group arranged and recorded two cover versions a year from 2008 to 2013, including
Caroline_Polachek
Type of vector space
Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler
Hecke_algebra
Non-singular cubic surface in mathematics
Hilbert modular surface of the level 2 principal congruence subgroup of the Hilbert modular group of the field Q(√5). The quotient of the Hilbert modular group
Clebsch_surface
Bulgarian mathematician (born 1986)
Swinnerton-Dyer: if a modular form f ( τ ) {\displaystyle f(\tau )} is not modular for some congruence subgroup of the modular group, then the Fourier
Vesselin_Dimitrov
Mathematical group of the homotopy classes of loops in a topological space
{\displaystyle \Gamma } any torsion-free congruence subgroup of the modular group SL ( 2 , Z ) {\displaystyle {\text{SL}}(2,\mathbb {Z} )} . From the
Fundamental_group
German mathematician (1849–1925)
algebra; Group theory; Geometry in more than 3 dimensions and differential equations, especially equations he invented, satisfied by elliptic modular functions
Felix_Klein
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
\Gamma } . In contrast to modular forms, Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949. The group G := S L 2 ( R ) = {
Maass_wave_form
Algebraic object
the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is the graded ring generated by the modular forms of Γ. The
Ring_of_modular_forms
Topics referred to by the same term
Iwasawa group or modular group, in the study of subgroup lattices, a group whose subgroup lattice is modular virtually polycyclic group, in infinite group theory
M-group
Number, approximately 3.14
properties under the modular group S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )} (or its various subgroups), a lattice in the group S L 2 ( R )
Pi
Type of algebraic equation
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problems. That is, given a number of functions
Modular_equation
Degree to which a system's components may be separated and recombined
Modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The
Modularity
Group of flat spacetime symmetries
The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It
Poincaré_group
Concept in modular arithmetic
In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent
Modular multiplicative inverse
Modular_multiplicative_inverse
Mathematical concept
complex modular forms and the p-adic theory of modular forms. Modular forms are analytic functions, so they admit a Fourier series. As modular forms also
Modular_forms_modulo_p
Series of Lego sets that resemble real life buildings
Lego Modular Buildings (stylized as LEGO Modular Buildings) is a series of Lego building toy sets introduced in 2007, with new sets usually being released
Lego_Modular_Buildings
modularity theorem is a theorem about modular tensor categories. It asserts that two different formulations of the modularity condition of a modular tensor
Bruguières_modularity_theorem
Modular form
A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient a0 in the Fourier series
Cusp_form
Topic in group theory and harmonic analysis (Niemeier lattice-mock theta connection)
Mathieu moonshine, where families of mock modular forms were attached to divisors of 24. After some group-theoretic discussion with Glauberman, Cheng
Umbral_moonshine
Part of the theory of modular forms
the modular group, with N ordered by divisibility. That is, if M divides N, Γ0(N) is a subgroup of Γ0(M). The oldforms for Γ0(N) are those modular forms
Atkin–Lehner_theory
Russian mathematician (born 1954)
mathematician who works on topology, geometry and group theory (particularly, modular Teichmüller groups). He is a professor at Michigan State University
Nikolai Ivanov (mathematician)
Nikolai_Ivanov_(mathematician)
Major type of automorphic form in mathematics
In mathematics, Siegel modular forms are a type of automorphic form that generalize conventional elliptic modular forms, which are closely related to elliptic
Siegel_modular_form
C) action. The action of SL(2, Z) is then the usual action of the modular group on the upper half-plane. Consequently, the period domain is the Riemann
Period_mapping
MODULAR GROUP
MODULAR GROUP
Girl/Female
Indian
Popular
Boy/Male
Arabic, Muslim
Famous; Popular
Boy/Male
Indian
Accepted, Popular
Boy/Male
Tamil
Parishrut | பரீஷà¯à®°à¯à®¤
Popular, Renown
Parishrut | பரீஷà¯à®°à¯à®¤
Boy/Male
Indian
Love
Boy/Male
Arabic, Muslim
Familiar; Popular
Girl/Female
Bengali, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Famous; Popular
Boy/Male
Muslim
Accepted, Popular
Boy/Male
Hindu
Popular, Renown
Boy/Male
Arabic
Popular; Famous
Girl/Female
Hindu, Indian
Popular Around
Boy/Male
Muslim
Accepted, Popular
Boy/Male
Muslim
Familiar, Popular
Girl/Female
Biblical Greek
Popular.
Girl/Female
Greek
Popular.
Girl/Female
Tamil
Popular
Girl/Female
Indian
Popular
Boy/Male
Arabic, Hindu, Indian, Muslim
Accepted; Popular
Boy/Male
Muslim/Islamic
Popular
Boy/Male
Hindu, Indian
Popular
MODULAR GROUP
MODULAR GROUP
Boy/Male
Hindu, Indian, Marathi, Tamil
God Sivan
Biblical
east; brightness
Female
Chinese
obliging and quiet.
Boy/Male
Indian, Sanskrit
Fire Tree; The Indian Redwood Tree
Boy/Male
Arabic, Muslim
Another Name for God; One who Creates
Surname or Lastname
English and Scottish
English and Scottish : habitational name from any of various minor places named Woodhead, for example in West Yorkshire and Strathmore, from Old English wudu ‘wood’ + hēafod ‘head(land)’, ‘top’, ‘extremity’.
Girl/Female
Indian, Tamil
Tasty
Biblical
uncomparable beauty
Boy/Male
Hindu, Indian, Tamil, Telugu
Goddess Lakshmi's Husband; Helper
Girl/Female
Tamil
Goddess Saraswati
MODULAR GROUP
MODULAR GROUP
MODULAR GROUP
MODULAR GROUP
MODULAR GROUP
a.
Prevailing among the people; epidemic; as, a popular disease.
a.
Adapted to the means of the common people; possessed or obtainable by the many; hence, cheap; common; ordinary; inferior; as, popular prices; popular amusements.
pl.
of Morula
p. pr. & vb. n.
of Modulate
a.
Of, pertaining to, or in the form of, a nodule or knot.
a.
Relating or belonging to an ovule; as, an ovular growth.
a.
Beloved or approved by the people; pleasing to people in general, or to many people; as, a popular preacher; a popular law; a popular administration.
n.
To model; also, to modulate.
pl.
of Modulus
a.
Of or pertaining to the common people, or to the whole body of the people, as distinguished from a select portion; as, the popular voice; popular elections.
a.
Of or pertaining to mode, modulation, module, or modius; as, modular arrangement; modular accent; modular measure.
a.
Depending on, or perceived by, the eye; received by actual sight; personally seeing or having seen; as, ocular proof.
v. t.
To vary or inflect in a natural, customary, or musical manner; as, the organs of speech modulate the voice in reading or speaking.
n.
The size of some one part, as the diameter of semi-diameter of the base of a shaft, taken as a unit of measure by which the proportions of the other parts of the composition are regulated. Generally, for columns, the semi-diameter is taken, and divided into a certain number of parts, called minutes (see Minute), though often the diameter is taken, and any dimension is said to be so many modules and minutes in height, breadth, or projection.
n.
A popular or jocular name for a drinking vessel.
a.
Given to jesting; jocose; as, a jocular person.
n.
Any one of the teeth back of the incisors and canines. The molar which replace the deciduous or milk teeth are designated as premolars, and those which are not preceded by deciduous teeth are sometimes called true molars. See Tooth.
a.
Having power to grind; grinding; as, the molar teeth; also, of or pertaining to the molar teeth.
a.
Popular; famous.
imp. & p. p.
of Modulate