ACTA ARITHMETICA 171.3 (2015) Cohen–Kuznetsov liftings of quasimodular forms by Min Ho Lee (Cedar Falls, IA) 1. Introduction. Quasimodular forms generalize classical modular forms, and they were introduced by Kaneko and Zagier in [3]. For a discrete subgroup Γ of SL(2, R) commensurable with SL(2, Z) and for nonnegative integers m and λ, a quasimodular form φ of weight λ and depth at most m for Γ corresponds to holomorphic functions φ 0 ,φ 1 ,...,φ m on the Poincar´ e upper half-plane H satisfying 1 (cz + d) λ φ  az + b cz + d  = φ 0 (z )+ φ 1 (z )  c cz + d  + ··· + φ m (z )  c cz + d  m for all z ∈H and ( ab cd ) ∈ Γ . The functions φ k are also quasimodular forms and are determined uniquely by φ. Thus φ determines the corresponding polynomial Φ(z,X )= m X r=0 φ r (z )X r of degree at most m in X . Such a polynomial is called quasimodular, and to study various aspects of quasimodular forms it is often convenient to work with quasimodular polynomials. Jacobi-like forms for Γ are formal power series which generalize Jacobi forms, and they were studied by Cohen, Manin and Zagier [2], [6]. It is known that there is a one-to-one correspondence between Jacobi-like forms and certain sequences of modular forms. In particular, for a modular form f , there is a Jacobi-like form e f (z,X ) corresponding to the sequence whose only nonzero term is f , which is known as the Cohen–Kuznetsov lifting of f . Although the coefficient functions of a Jacobi-like form are not modular forms in general, they are in fact quasimodular forms. There is a surjective map from the space of Jacobi-like forms to the space of quasimodular poly- 2010 Mathematics Subject Classification : 11F11, 11F50. Key words and phrases : Cohen–Kuznetsov liftings, quasimodular forms, Jacobi-like forms, modular forms. DOI: 10.4064/aa171-3-3 [241] c  Instytut Matematyczny PAN, 2015