Quasimodular forms and automorphic pseudodifferential operators of mixed weight
Jacobi-like forms for a discrete subgroup \(\Gamma \) of \(SL(2, \mathbb R)\) are formal power series which generalize Jacobi forms, and they are in one-to-one correspondence with automorphic pseudodifferential operators for \(\Gamma \). The well-known Cohen–Kuznetsov lifting of a modular form f provides a Jacobi-like form and therefore an automorphic pseudodifferential operator associated to f. Given a pair \((\lambda , \mu )\) of integers, automorphic pseudodifferential operators can be extended to those of mixed weight. We show that each coefficient of an automorphic pseudodifferential operator of mixed weight is a quasimodular form and prove the existence of a lifting of Cohen–Kuznetsov type for each quasimodular form.
KeywordsQuasimodular forms Modular forms Automorphic pseudodifferential operators Jacobi-like forms
Mathematics Subject Classification11F11 11F99
The author would like to thank the referee for various comments and suggestions.
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