harmonic Maass forms

harmonic Maass forms
Name	harmonic Maass forms
Field	Number theory
Introduced	1930s–2000s
Notable	Hans Maass, Sander Zwegers, Don Zagier

harmonic Maass forms are certain real-analytic functions on the upper half-plane that transform like modular forms under the action of discrete subgroups of SL(2, Z), satisfy a growth condition at cusps, and are annihilated by the hyperbolic Laplacian. They generalize classical Modular forms and provide the analytic framework linking Ramanujan's mock theta functions, the theory of Hecke operators, and arithmetic invariants arising in Elliptic curve and K3 surface contexts.

Definition and basic properties

A harmonic Maass form for a congruence subgroup such as Γ0(N), Γ1(N), or SL(2, Z) is a smooth function on the upper half-plane that transforms with a given weight and multiplier system under the action of matrices from SL(2, Z), is annihilated by the weight-k hyperbolic Laplacian Δ_k, and has at most linear exponential growth at all cusps including the cusp at infinity. Foundational contributors include Hans Maass, Jean-Pierre Serre, and Atkin–Lehner theory figures; modern development owes much to Ken Ono, Don Zagier, Sander Zwegers, and Jan Bruinier. Key structural properties involve spectral decomposition relative to the Laplacian (operator), pairing with cuspidal Hecke eigenforms, and compatibility with Petersson inner products in regularized forms.

Examples and canonical constructions

Canonical examples arise from Poincaré series built from seed functions invariant under parabolic subgroups and from nonholomorphic Eisenstein series attached to Dirichlet characters and Dedekind eta function variants. Specific constructions include harmonic lifts of classical Eisenstein series studied by Don Zagier and harmonic Maass forms associated to weight-1/2 theta series connected to Shimura correspondence and Shintani lift phenomena. Explicit forms appear in work of Sander Zwegers on Ramanujan's mock theta functions, in Bruinier–Funke theta lifts involving Borcherds products, and in the arithmetic theta lift framework developed by Stephen Kudla and John Millson.

Fourier expansions and principal parts

At each cusp a harmonic Maass form admits a Fourier expansion splitting into a holomorphic part and a nonholomorphic part; the holomorphic part often has a principal part given by a finite polar part, while the nonholomorphic part involves incomplete gamma functions and period integrals of weight-2−k modular forms. The principal part encodes arithmetic data such as Fourier coefficients connected to partition function congruences investigated by Ramanujan and G. H. Hardy, and coefficients related to central values of L-functions tied to Birch and Swinnerton-Dyer conjecture instances. Analysis of Fourier coefficients uses spectral theory from Selberg trace formula, analytic continuation methods from Hecke theory, and explicit expansions found in the work of Bringmann and Ono.

Differential operators and exact sequences

Important operators include the lowering operator ξ_k mapping harmonic Maass forms of weight k to cusp forms of weight 2−k, and the Maass raising and lowering operators relating different weights; these induce exact sequences linking spaces of weakly holomorphic modular forms, harmonic Maass forms, and classical cusp forms. The ξ-operator was central to breakthroughs by Bruinier and Funke, enabling the construction of Green functions on arithmetic quotients and connections to Arakelov theory and Gross–Zagier formula settings. Cohomological frameworks draw on ideas from Deligne cohomology, Eichler cohomology, and the Hodge theory of modular curves such as X0(N).

Connections to modular forms and mock modular forms

Harmonic Maass forms provide the natural home for the holomorphic parts known as mock modular forms; this relation was clarified by Sander Zwegers who showed that adding a nonholomorphic correction term yields modularity. Through the Shimura correspondence, harmonic Maass forms of half-integral weight link to integral-weight forms, while the Bruinier–Funke pairing relates their shadows to cusp forms studied by Pieter Moree and Nicolas Templier. The interplay underlies work on Moonshine phenomena connecting finite groups like the Monster (group) and modular objects studied by John Conway and Simon Norton.

Applications in number theory and geometry

Applications span arithmetic and geometric domains: coefficients of harmonic Maass forms encode partition asymptotics and congruences related to Ramanujan tau function investigations, while theta-lift constructions produce automorphic Green functions and divisors on moduli spaces such as those studied in Kudla's Program and Arakelov geometry. They play roles in explicit formulas for heights on Elliptic curves in the context of Gross–Zagier formula and in counts of rational points on K3 surfaces and Calabi–Yau manifold moduli via Borcherds products. Interactions with analytic number theory involve central values of L-functions, nonvanishing results tied to Wiles and Taylor–Wiles method contexts, and relations to quantum invariants appearing in Chern–Simons theory and Topological quantum field theory settings.

Category:Harmonic analysis