Weakly holomorphic modular form

Template:Short description Template:Distinguish 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 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

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