In mathematics, a modular form is a holomorphic function on the complex upper half-plane, H {\displaystyle {\mathcal {H}}} , that roughly satisfies a functional...

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...

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...

In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related...

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...

Teichmüller modular form is an analogue of a Siegel modular form on Teichmüller space. Ichikawa, Takashi (1994), "On Teichmüller modular forms", Mathematische...

In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer...

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...

announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations". However, in September...

Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic...

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...

from modular arithmetic. The modular group Γ is the group of fractional linear transformations of the complex upper half-plane, which have the form z ↦...

In mathematics, overconvergent modular forms are special p-adic modular forms that are elements of certain p-adic Banach spaces (usually infinite dimensional)...

mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms that are polynomials in 1/Im(τ)...

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 synthesizers are synthesizers composed of separate modules for different functions. The modules can be connected together by the user to create...

mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. A cusp form is distinguished...

number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way...

holomorphic modular form is similar to a holomorphic modular form, except that it is allowed to have poles at cusps. Examples include modular functions...

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...

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...

Maass form is a smooth function f {\displaystyle f} on the upper half plane, transforming like a modular form under the action of the modular group,...

introduced by Petersson (1930), is a generalization to other modular forms or automorphic forms. The Riemann zeta function and the Dirichlet L-function satisfy...

In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Erich Hecke (1937a,1937b), is a certain kind of "averaging"...

generating function as the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence...

Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as...

A modular elliptic curve is an elliptic curve E that admits a parametrization X0(N) → E by a modular curve. This is not the same as a modular curve that...

lattice is the underlying object with which elliptic functions and modular forms are defined. A fundamental pair of periods is a pair of complex numbers...

between modular forms and elliptic curves. In the 1950s post-World War II period of mathematics, there was renewed interest in the theory of modular curves...

}^{\infty }e^{2\pi inz\ +\ \pi in^{2}\tau }} which is a kind of modular form called a Jacobi form. This is sometimes written in terms of the nome q = e π i...