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.

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 (which generate the ring of holomorphic modular forms) 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.