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Quasimodular forms and Jacobi-like forms. (English) Zbl 1327.11027
In the paper under consideration, the authors provide many interesting theorems involving the construction and properties of various maps between spaces of modular, quasimodular, and Jacobi-like forms.
One of the interesting constructions which aids the authors’ constructions is the passage from quasimodular forms to quasimodular polynomials. This builds a polynomial in a natural way out of the pieces arising in the error to modularity of a quasimoudular form (see page 644). In (2.7), the authors give a natural action of \(\operatorname{GL}_2(\mathbb R)^+\) on such polynomials, and the space of polynomials which are invariant under this action is shown to be isomorphic to the space of quasimodular forms in Theorem 3.4. Armed with this perspecitive, the authors proceed to analyze various maps between spaces of quasimodular forms and Jacobi-like forms, which play a key role in, for example, the Cohen-Kusnetzov lifting. In Proposition 3.10, they provide a lifting from Jacobi-like forms to quasimodular polynomials. Moreover, they use the slash operators which characterize these two objects to give a natural definition of Hecke operators on both spaces, and show the Hecke-equivariance of this map in Theorem 4.5.
They also describe an interesting lifting from modular to quasimodular forms in Proposition 5.1, a lifting from quasi-modular forms to Jacobi-like forms in Corollary 6.3, as well as a number of other interesting maps, showing Hecke-equivariance of the key liftings. The reader will also enjoy illuminating examples given in Remark 6.4, as well as in Examples 6.5, 6.6.

MSC:
11F11 Holomorphic modular forms of integral weight
11F50 Jacobi forms
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