Dedekind sum
In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation
Modular forms modulo p
In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime p, there is an analogous
Harmonic Maass form
In mathematics, a weak Maass form is a smooth function on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hype
Q-expansion principle
In mathematics, the q-expansion principle states that a modular form f has coefficients in a module M if its q-expansion at enough cusps resembles the q-expansion of a modular form g with coefficients
Ramanujan tau function
The Ramanujan tau function, studied by Ramanujan, is the function defined by the following identity: where q = exp(2πiz) with Im z > 0, is the Euler function, η is the Dedekind eta function, and the f
Quantum ergodicity
In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the quantization of classical mechanical systems that are chaotic in the sense of exponential sensitivity to ini
Modular unit
In mathematics, modular units are certain units of rings of integers of fields of modular functions, introduced by Kubert and Lang. They are functions whose zeroes and poles are confined to the cusps
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 of the modular group, and also
Picard–Fuchs equation
In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves.
Shimura subgroup
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 to the Jacobian of X1(N). It is named after Goro S
Theta constant
In mathematics, a theta constant or Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction θm(τ) = θm(τ,0) of a theta function θm(τ,z) with rational characteristic m to
Automorphic factor
In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is review
Waldspurger's theorem
In mathematics, Waldspurger's theorem, introduced by Jean-Loup Waldspurger, is a result that identifies Fourier coefficients of modular forms of half-integral weight k+1/2 with the value of an L-serie
Weierstrass elliptic function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-f
Eisenstein ideal
In mathematics, the Eisenstein ideal is an ideal in the endomorphism ring of the Jacobian variety of a modular curve, consisting roughly of elements of the Hecke algebra of Hecke operators that annihi
Doi–Naganuma lifting
In mathematics, the Doi–Naganuma lifting is a map from elliptic modular forms to Hilbert modular forms of a real quadratic field, introduced by and .It was a precursor of the base change lifting. It i
P-adic modular form
In mathematics, a p-adic modular form is a p-adic analog of a modular form, with coefficients that are p-adic numbers rather than complex numbers. introduced p-adic modular forms as limits of ordinary
Lemniscate elliptic functions
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonha
Ikeda lift
In mathematics, the Ikeda lift is a lifting of modular forms to Siegel modular forms. The existence of the lifting was conjectured by W. Duke and Ö. Imamoḡlu and also by T. Ibukiyama, and the lifting
Petersson inner product
In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson.
Serre's modularity conjecture
In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form.
Fundamental pair of periods
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functi
Weber modular function
In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber.
Upper half-plane
In mathematics, the upper half-plane, is the set of points (x, y) in the Cartesian plane with y > 0.
Schwarzian derivative
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in
Hard hexagon model
In statistical mechanics, the hard hexagon model is a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular lattice but no two particles may be adjace
Jacobi group
In mathematics, the Jacobi group, introduced by , is the semidirect product of the symplectic group Sp2n(R) and the Heisenberg group R1+2n. The concept is named after Carl Gustav Jacob Jacobi. Automor
Ramanujan–Petersson conjecture
In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan , states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12 where , satisfies
Eichler–Shimura isomorphism
In mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for Fuchsian groups, introduced by Eichler, that is a variation of group cohomology
Eigencurve
In number theory, an eigencurve is a rigid analytic curve that parametrizes certain p-adic families of modular forms, and an eigenvariety is a higher-dimensional generalization of this. Eigencurves we
Overconvergent modular form
In mathematics, overconvergent modular forms are special p-adic modular forms that are elements of certain p-adic Banach spaces (usually infinite dimensional) containing classical spaces of modular fo
Poincaré series (modular form)
In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several com
Atkin–Lehner theory
In mathematics, Atkin–Lehner theory is part of the theory of modular forms describing when they arise at a given integer level N in such a way that the theory of Hecke operators can be extended to hig
J-invariant
In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex nu
Real analytic Eisenstein series
In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely rel
Equianharmonic
In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g2 = 0 and g3 = 1.This page follows the terminolog
Schwarz triangle function
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs fo
Umbral moonshine
In mathematics, umbral moonshine is a mysterious connection between Niemeier lattices and Ramanujan's mock theta functions. It is a generalization of the Mathieu moonshine phenomenon connecting repres
Ribet's theorem
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jea
Shimura correspondence
In number theory, the Shimura correspondence is a correspondence between modular forms F of half integral weight k+1/2, and modular forms f of even weight 2k, discovered by Goro Shimura. It has the pr
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds cons
Maass–Selberg relations
In mathematics, the Maass–Selberg relations are some relations describing the inner products of truncated real analytic Eisenstein series, that in some sense say that distinct Eisenstein series are or
Weakly holomorphic modular form
In mathematics, a weakly holomorphic modular form is similar to a holomorphic modular form, except that it is allowed to have poles at cusps. Examples include modular functions and modular forms.
Modular lambda function
In mathematics, the modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group
Elliptic cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
Saito–Kurokawa lift
In mathematics, the Saito–Kurokawa lift (or lifting) takes elliptic modular forms to Siegel modular forms of degree 2. The existence of this lifting was conjectured in 1977 independently by Hiroshi Sa
Diamond operator
In number theory, the diamond operators 〈d〉 are operators acting on the space of modular forms for the group Γ1(N), given by the action of a matrix (a bc δ) in Γ0(N) where δ ≈ d mod N. The diamond ope
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
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 matr
Cusp form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion.
Thomae's formula
In mathematics, Thomae's formula is a formula introduced by Carl Johannes Thomae relating theta constants to the branch points of a hyperelliptic curve .
Classical modular 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)) is a point on the curve. Here j(τ) denotes the
Rogers–Ramanujan identities
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Roger
Frobenius solution to the hypergeometric equation
In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that us
Dedekind eta function
In 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 complex numbers, where the imaginary part
Elliptic unit
In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. T
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators.
Kronecker limit formula
In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There
Congruence ideal
In algebra, the congruence ideal of a surjective ring homomorphism f : B → C of commutative rings is the image under f of the annihilator of the kernel of f. It is called a congruence ideal because wh
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modu
Mock modular form
In 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 1/2. The first examples of mock theta
Manin–Drinfeld theorem
In mathematics, the Manin–Drinfeld theorem, proved by Manin and Drinfeld, states that the difference of two cusps of a modular curve has finite order in the Jacobian variety.
Fricke involution
In mathematics, a Fricke involution is the involution of the modular curve X0(N) given by τ → –1/Nτ. It is named after Robert Fricke. The Fricke involution also acts on other objects associated with t
Modular group
In mathematics, the modular group is the projective special linear group PSL(2, Z) of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular grou
Jacobi form
In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group . The theory was first systematically
Almost holomorphic modular form
In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms that are polynomials in 1/Im(τ) with coefficients that are holomor
Modular symbol
In mathematics, modular symbols, introduced independently by Bryan John Birch and by Manin, span a vector space closely related to a space of modular forms, on which the action of the Hecke algebra ca
Rogers–Ramanujan continued fraction
The Rogers–Ramanujan continued fraction is a continued fraction discovered by and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated expl
Drinfeld reciprocity
In mathematics, Drinfeld reciprocity, introduced by Drinfeld, is a correspondence between eigenforms of the moduli space of Drinfeld modules and factors of the corresponding Jacobian variety, such tha
Eichler–Shimura congruence relation
In number theory, the Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators. It was introduced by Eichler an
Rankin–Cohen bracket
In mathematics, the Rankin–Cohen bracket of two modular forms is another modular form, generalizing the product of two modular forms.Rankin gave some general conditions for polynomials in derivatives
Paramodular group
In mathematics, 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 relation to polarized abelian
Ring of modular forms
In mathematics, 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 study of rings of modular forms de
Hecke operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Erich Hecke, is a certain kind of "averaging" operator that plays a significant role in the structure of vect
Eigenform
In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form which is an eigenvector for all Hecke operators Tm, m = 1, 2, 3, .... Eigenforms fall i
Modularity theorem
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational n
Modular 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 on a moduli space, a modular equation is an equat