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.

Not to be confused with almost holomorphic modular form or nearly holomorphic modular form.

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: ${\displaystyle f((a\tau +b)/(c\tau +d))=(c\tau +d)^{k}f(\tau )}$ 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 whose radius of convergence is 1 (so f is holomorphic on the upper half plane and meromorphic at the 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.

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.

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, doi:10.4310/PAMQ.2008.v4.n4.a15, MR 2441704, Zbl 1200.11027