- modular form: (1) modularity, (2) holomorphic, (3) holomorphic at cusps.

How they are generalized or restricted ?

- cusp form: (3) vanishes at cusps.
- non-holomoprhic modular form(?): (2) real analytic, and eigenfunction of Casimir operator.
- weakly holomorphic modular form: (3) allowing finite-order pole at cusps.
- harmonic Maass form: (2) real analytic, and annihilated by weight-$k$-Laplacian. (3) at most linear exponential growth at cusps.
- weak Maass form: (2) real analytic, and eigenfunction of weight-$k$-Laplacian. (3) at most linear exponential growth at cusps.
- mock modular form: ``holomorphic part'' of harmonic Maass form.
- meromorphic modular form:
- weakly holomorphic cusp form: weakly holomorphic, and its Fourier expansion has no $q^0$ terms. (still may allow $q^n$ with negative $n$.)
- quasi-modular form

Generalization to another direction

- modular form with half integral weight
- Siegel modular form
- Jacobi form

Notations

- $M_k$ modular form
- $S_k$ cusp form
- $M^!_k$ weakly holomorphic modular form
- $S^!_k$ weakly holomorphic cusp form
- $QM_k$ quasi-modular form

Last-modified: 2016-11-24 (木) 15:11:20