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Modular forms and differential operators. (English) Zbl 0806.11022
The author investigates the general background of the effect of differential operators in the theory of modular forms. Following the work of Rankin and Cohen one can assign to modular forms \(f\) and \(g\) of weight \(k\) and \(\ell\) a modular form \([f,g]_ n\) of weight \(k+\ell+2n\). Various algebraic relations among these Rankin-Cohen brackets are derived. Connections with Jacobi forms and pseudodifferential operators are given. On the other hand these algebraic relations are the basis of the definition of an abstract algebraic structure, the so-called RC algebra. Under certain conditions the RC-algebra becomes equivalent to a commutative graded algebra endowed with a derivation of degree 2 and a particular element of degree 4.
Reviewer: A.Krieg (Aachen)

MSC:
11F11 Holomorphic modular forms of integral weight
35S99 Pseudodifferential operators and other generalizations of partial differential operators
17B69 Vertex operators; vertex operator algebras and related structures
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