modulform

Modulform, often referred to in English as a modular form, denotes a class of complex analytic functions on the upper half-plane that satisfy a specific transformation behavior under the action of a subgroup of SL(2,Z). Each modulform is holomorphic on the upper half-plane and at the cusps, and it may be further classified as a cusp form if it vanishes at all cusps; otherwise it is a non-cuspidal modular form. The subject is central to modern number theory and has deep connections to arithmetic, geometry, and mathematical physics.

A modulform f is typically described by three parameters: weight k, level N, and a Dirichlet character χ modulo N. For the full modular group SL(2,Z) the defining transformation law is f((az+b)/(cz+d)) = (cz+d)^k f(z) for all integers a, b, c, d with ad−bc=1. For congruence subgroups such as Γ0(N), the law generalizes to f(γz) = χ(d)(cz+d)^k f(z) for γ in Γ0(N). The Fourier expansion at infinity, written f(z) = ∑_{n≥0} a_n q^n with q = e^{2πi z}, encodes arithmetic information in the coefficients a_n.

The space M_k(Γ, χ) of modulforms includes subspaces such as cusp forms S_k(Γ, χ). Hecke operators T_n act on these spaces, commuting with each other, and enabling the study of eigenforms and associated L-functions via Euler products. Classic examples include the cusp form Δ(z) of weight 12 for SL(2,Z) and the Eisenstein series E_k, which generate many other forms and relations, such as the modular j-invariant constructed from Eisenstein series and the discriminant.

Modulforms have broad applications, including the modularity theorem for elliptic curves, arithmetic geometry, partition theory, and connections to moonshine phenomena. The term modulform is used in some languages as the equivalent of the English “modular form.”