Quasimodular forms and quasimodular polynomials (Q1933467)

In this nice expository article, the author provides some recent developments in the theory of quasi-modular forms. Let \(\Gamma\) be a discrete subgroup of \(SL_2(\mathbb R)\). First, the author describes the relations between quasi modular forms, quasi modular polynomials and modular polynomials for \(\Gamma\). Then the author identifies quasi-modular forms for \(\Gamma\) as sections of some vector bundles over the quotient space \(\Gamma \setminus \mathcal{H}\). This generalises the notion of modular forms for \(\Gamma\) as holomorphic sections of line bundles over \(\Gamma \setminus \mathcal{H}\). Next the author discusses the connections of quasi-modular polynomials with Jacobi-like forms. Finally, the author describes a Hecke equivariant linear map from quasi modular forms for \(\Gamma\) to cohomology classes of \(\Gamma\).