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| mock modular forms | |
|---|---|
| Name | Mock modular forms |
| Field | Mathematics |
| Introduced | Early 20th century; revival in 21st century |
| Notable | Srinivasa Ramanujan; Sander Zwegers; Don Zagier; Ken Ono |
mock modular forms are a class of holomorphic q-series that exhibit modular-like transformation behavior but are not modular forms. They arise from early 20th-century work by Srinivasa Ramanujan and were systematically developed in the late 20th and early 21st centuries by researchers such as Sander Zwegers, Don Zagier, and Ken Ono. Mock modular forms connect to diverse parts of mathematics and physics including the theory of modular forms, Maass wave forms, elliptic curves, Moonshine (mathematics), and string theory.
A mock modular form is a holomorphic function on the upper half-plane whose nonholomorphic completion transforms like a modular form for a subgroup of SL(2,Z), often with specified weight and multiplier. Key properties include q-expansions at cusps, growth conditions related to Fourier coefficients studied by G.H. Hardy, Ramanujan, Atkin, and Hecke, and connections to principal parts determined by poles at cusps as in the work of André Weil and Erich Hecke. Mock modular forms often pair with a "shadow" that is a cusp form studied in the context of Petersson inner product and Hecke operators by researchers such as Jacques Hadamard and Atle Selberg.
Ramanujan introduced "mock theta functions" in letters to G.H. Hardy and in his last letter to Godfrey Harold Hardy and unpublished notebooks, which puzzled analysts including B. M. Wilson and Bruce C. Berndt. The phenomenon remained mysterious until Sander Zwegers connected Ramanujan's mock theta functions to nonholomorphic completions using ideas from Harmonic analysis and the theory of Lorentzian lattices inspired by work of Venkov and Borcherds. Subsequent advances came from Don Zagier, Zagier's Eichler integral theory, Ken Ono, Kathrin Bringmann, Jan Bruinier, and Nick Andersen, integrating techniques from the Langlands program and the study of automorphic forms by figures like Robert Langlands and Harish-Chandra.
Classical examples include Ramanujan's mock theta functions such as the fifth-order mock theta functions; specific q-series were studied by Ramanujan, George Andrews, Frank Garvan, Fredrik Johansson, David Boyd, and Iain Gordon. Modern constructions produce mock modular forms from isolating holomorphic parts of harmonic Maass forms following approaches by Sander Zwegers, Kathrin Bringmann, Ken Ono, Jan Bruinier, and Olav Richter. Other constructions arise via theta lifts and regularized theta integrals developed by Borcherds, Stephen Kudla, John Millson, Stephen S. Gelbart, Ralph Greenberg, and Shamit Kachru in contexts overlapping with Calabi–Yau manifolds studied by Maxim Kontsevich.
Every mock modular form has an associated "shadow", typically a cusp form or theta function that governs its nonholomorphic completion; this concept was clarified by Sander Zwegers and formalized by Don Zagier. Completions transform under SL(2,Z), often requiring multiplier systems tied to characters like those studied by Hecke and Dirichlet; analyses of shadows use tools from Poincaré series examined by Henri Poincaré and spectral theory of the Laplace operator influenced by Atle Selberg and Ilya Piatetski-Shapiro.
Mock modular forms are the holomorphic parts of harmonic Maass forms of weight k, a viewpoint developed by Jan Bruinier, Ken Ono, Kathrin Bringmann, Olav Richter, and Sander Zwegers. Harmonic Maass forms generalize classical modular forms studied by Erich Hecke and Goro Shimura; they decompose into holomorphic mock parts and nonholomorphic shadows, linking to the Eichler–Shimura isomorphism investigated by Don Zagier and Goro Shimura and to cohomological perspectives advanced by Pierre Deligne and Jean-Pierre Serre.
Mock modular forms appear in partition theory through work of Ramanujan, George Andrews, Frank Garvan, and Ken Ono connecting to congruences of p(n) and to ranks and cranks studied by Freeman Dyson. They also surface in moonshine phenomena linking finite groups like the Monster group, investigated by John Conway, Simon Norton, and Richard Borcherds, to q-series resembling mock modular forms. In physics, mock modular forms arise in black hole entropy counts, BPS state enumerations, and topological string theory studied by Strominger, Vafa, Ashoke Sen, Gaberdiel, and Tuite, as well as in the study of vertex operator algebras by Frenkel, Lepowsky, and Meurman.
Explicit computations use q-series manipulations, Rademacher-type sums developed from Hans Rademacher, holomorphic projection methods employed by Kohnen and Zagier, and theta lift algorithms advanced by Jan Bruinier and Stephen Kudla. Software implementations leverage libraries developed by contributors like Fredrik Johansson, William Stein, John Cremona, and Brendan McKay to compute Fourier coefficients, modular transformations, and shadows; high-precision computations apply methods from Andreas Enge and Antoine Joux in computational number theory.
Active research addresses classification of mock modular forms for congruence subgroups examined by Andrew Wiles and Richard Taylor, connections with the Langlands program pursued by Robert Langlands and Edward Frenkel, deeper moonshine correspondences explored by John Duncan and Miranda Cheng, and arithmetic applications to ranks of elliptic curves pioneered by Birch and Swinnerton-Dyer and Freeman Dyson. Other directions include categorification approaches linked to Mikhail Khovanov and Jacob Lurie, physical realizations in AdS/CFT correspondence studied by Juan Maldacena, and effective bounds on coefficients using analytic techniques of Iwaniec, Henryk Iwaniec, and Peter Sarnak.