maass

Maass is a German surname. In mathematics, Maass form refers to a class of automorphic forms introduced by the German mathematician Hans Maass. Maass forms are real-analytic functions on the upper half-plane that are invariant under the action of a discrete subgroup Γ of SL(2,R), typically Γ = SL(2,Z). They are eigenfunctions of the hyperbolic Laplacian and are not, in general, holomorphic.

Concretely, if z = x + i y lies in the upper half-plane, a Maass form f satisfies f(γz) = f(z) for all γ in Γ, is an eigenfunction of Δ = -y^2(∂^2/∂x^2 + ∂^2/∂y^2) with eigenvalue λ, and exhibits at most exponential growth toward cusps. Maass cusp forms are those Maass forms that vanish at all cusps; otherwise they are ordinary Maass forms. Their Fourier expansions at the cusps involve K-Bessel functions and a continuous spectral parameter related to λ (often written as λ = 1/4 + t^2).

Maass forms are central to the spectral theory of automorphic forms on the modular group and to the study of L-functions and arithmetic through their Fourier coefficients. They serve as non-holomorphic counterparts to classical holomorphic modular forms and connect number theory with representation theory and mathematical physics.

Usage: The term Maass form also appears in discussions of Maass forms for other groups and in the broader study of automorphic forms.