Special functions
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Function (mathematics)
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Number theory
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Complex analysis
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Analytic number theory
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Moduli space
Automorphic form
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...
Elliptic function
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Monstrous moonshine
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Theta function
Elliptic function
In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an ellip...
Elliptic function - Wikipedia
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modul...
Monstrous moonshine - Wikipedia
Theta function
In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic fo...
Theta function - Wikipedia
Classical modular curve
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equationsuch that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here j(τ) denotes the j-invariant.Th...
Classical modular curve - Wikipedia
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...
Frobenius solution to the hypergeometric equation - Wikipedia
Tau-function
The Ramanujan tau function, studied by Ramanujan (1916), is the function defined by the following identity:where with and is the Dedekind eta function and the function is a holomorphic cusp...
Tau-function - Wikipedia
Theta divisor
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the asso...
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 funct...
Fundamental pair of periods - Wikipedia
Upper half-plane
In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part:The term arises from a common visualization of the complex number x + iy as the point (x,y) in the pla...
Eigenform
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 ...
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...
Hecke operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Hecke (1937), is a certain kind of "averaging" operator that plays a significant role in the structure o...
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...
Nome (mathematics)
In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK ′ are the quarter periods, and ω1 and ω2 are the fund...
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...
Eisenstein series - Wikipedia
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 ...
Ramanujan theta function
In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi trip...
Ramanujan theta function - Wikipedia
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...
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 f...
Ribet's theorem
In mathematics, Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is a statement in number theory concerning properties of Galois representations associated with modular forms. I...
Rogers–Ramanujan continued fraction
The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. I...
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 that annihilate the Eisenstein...
Monster Lie algebra
In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.
The mon...
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. Serre (1973) introduced p-adic modular forms as limit...
Jacobi group
In mathematics, the Jacobi group, introduced by Eichler & Zagier (1985), is the semidirect product of the symplectic group Sp2n(R) and the Heisenberg group R. The concept is named after Carl Gust...
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...
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 ...
Dedekind eta function - Wikipedia
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...
Modular curve - Wikipedia
Q-expansion principle
In mathematics, the q-expansion principle states that a modular form has coefficients in some module provided that its q-expansion at enough cusps resembles that of such a form. It was introduced by K...
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...