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:

f transforms like a modular form: $f((a\tau +b)/(c\tau +d))=(c\tau +d)^{k}f(\tau )$ for some integer k called the weight, for any elements of SL₂(Z).
As a function of q=e^2πiτ, f is given by a Laurent series (so it is allowed to have poles at cusps).

Examples

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.

References

Duke, W.; Jenkins, Paul (2008), "On the zeros and coefficients of certain weakly holomorphic modular forms", Pure Appl. Math. Q., Special Issue: In honor of Jean-Pierre Serre. Part 1, 4 (4): 1327–1340, MR 2441704