Modular form

A modular form is a holomorphic function on the upper half-plane that transforms predictably under a modular group and satisfies a condition at its cusps.

Modular group: A group of integer matrices acting on the upper half-plane by fractional linear transformations. Its transformations impose the symmetries that a modular form must obey.
Modular transformation: A fractional linear map induced by a matrix in the modular group. The defining law prescribes how a form changes under these maps.
Ramanujan tau function: The integer sequence given by the coefficients of the weight-12 cusp form Δ. Its coefficients exhibit striking divisibility and growth properties of modular forms.
Carl Gustav Jacob Jacobi: A nineteenth-century mathematician whose work on elliptic functions and theta functions shaped analysis. His theta-function identities supplied early examples of modular transformation behavior.
Modular function: A meromorphic function invariant under a modular group, with no nonzero weight factor. It has weight zero, unlike forms whose transformation law includes a weight factor.
Upper half-plane: The set of complex numbers with positive imaginary part. Modular forms are defined as holomorphic functions on this domain.
Slash operator: An operator describing the weight-dependent action of a matrix on a function of a complex variable. It expresses the transformation law compactly, including for non-invariant forms.
Modularity theorem: The theorem that every elliptic curve over the rational numbers corresponds to a modular form. It links modular forms to elliptic curves and enabled the proof of Fermat’s Last Theorem.
Évariste Galois: A French mathematician whose theory of equations introduced the group-theoretic framework now bearing his name. The modular group’s algebraic setting reflects the broader role of groups in symmetry.
Quasimodular form: A function whose modular transformation law includes a controlled polynomial correction term. It relaxes exact modularity while retaining structured transformation behavior.

Linked from 21 pages