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Quasimodular forms and quasimodular polynomials. (Formes quasimodulaires et polynômes quasimodulaires.) (English. French summary) Zbl 1283.11071
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 \(\Gamma\).
MSC:
11F11 Holomorphic modular forms of integral weight
11F23 Relations with algebraic geometry and topology
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