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

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

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...

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

rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic...

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

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

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...

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"...

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

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...

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

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...

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 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...

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

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

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...

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...

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

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

reliability of a system Indeterminate form, an algebraic expression that cannot be used to evaluate a limit Modular form, a (complex) analytic function on...

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...

modular discriminant is non-zero. This is due to the corresponding cubic polynomial having distinct roots. It can be shown that Δ is a modular form of...

and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura conjecture (eventually as the modularity theorem)...

are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein...

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, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the...

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...