Almost holomorphic modular form

In mathematics, almost holomorphic modular forms are a generalization of modular forms that are polynomials in 1/Im(τ) with coefficients that are holomorphic functions of τ. A quasimodular form is the holomorphic part of an almost holomorphic modular form. The operation of taking the holomorphic part gives an isomorphism between the spaces of almost holomorphic modular forms and quasimodular forms. The archetypal examples of quasi modular forms are the Eisenstein series E₂(τ) (the holomorphic part of the almost holomorphic modular form E₂(τ) – 3/πIm(τ)), and derivatives of modular forms.

In terms of representation theory, modular forms correspond roughly to highest weight vectors of certain discrete series representations of SL₂(R), while almost holomorphic or quasimodular forms correspond roughly to other (not necessarily highest weight) vectors of these representations.

Definitions

To simplify notation this section does the level 1 case; the extension to higher levels is straightforward.

A level 1 almost 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 a polynomial in 1/Im(τ) with coefficients that are holomorphic functions of q.

A level 1 quasimodular form is defined to be the constant term of an almost holomorphic modular form (considered as a polynomial in 1/Im(τ)).

Structure

The ring of almost holomorphic modular forms of level 1 is a polynomial ring over the complex numbers in the three generators $E_{2}(\tau )-3/\pi \Im (\tau ),E_{4}(\tau ),E_{6}(\tau )$. Similarly The ring of quasimodular forms of level 1 is a polynomial ring over the complex numbers in the three generators $E_{2}(\tau ),E_{4}(\tau ),E_{6}(\tau )$.

Derivatives

Ramanujan observed that the derivative of any quasimodular form is another quasimodular form. For example,

${\frac {dE_{2}}{d\tau }}={\frac {E_{2}^{2}-E_{4}}{12}}$

${\frac {dE_{4}}{d\tau }}={\frac {E_{2}E_{4}-E_{6}}{3}}$

${\frac {dE_{6}}{d\tau }}={\frac {E_{2}E_{6}-E_{4}^{2}}{2}}$

As the field generated by quasimodular forms of some level has transcendence degree 3 over C, this implies that any quasimodular form satisfies some nonlinear differential equation of order 3. For example, the Eisenstein series E₂ satisfies Chazy equation (give or take a few constants).

References

• Zagier, Don (2008), "Elliptic modular forms and their applications", The 1-2-3 of modular forms, Universitext, Berlin: Springer, pp. 1–103, doi:10.1007/978-3-540-74119-0, ISBN 978-3-540-74117-6, MR 2409678