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

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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:   for some integer k called the weight, for any elements of SL2(Z).
  • As a function of q=eiτ, 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

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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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  • 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