ochiai/mock

• 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