ochiai/mock - PukiWiki
  • 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

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Last-modified: 2016-11-24 (木) 15:11:20 (363d)