Weakly holomorphic modular form


In mathematics, a weakly holomorphic modular form is similar to a holomorphic modular form, except that it is allowed to have poles at cusps. Examples include modular functions and modular forms.

Definition

To simplify notation this section does the level 1 case; the extension to higher levels is straightforward.
A level 1 weakly holomorphic modular form is a function f on the upper half plane with the properties:
The ring of level 1 modular forms is generated by the Eisenstein series E4 and E6 together with the inverse 1/Δ of the modular discriminant.
Any weakly holomorphic modular form of any level can be written as a quotient of two holomorphic modular forms. However, not every quotient of two holomorphic modular forms is a weakly holomorphic modular form, as it may have poles in the upper half plane.