Modular forms and differential operators

Abstract

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for each n ≥ 0 a bilinear operation which assigns to two modular forms f and g of weight k and l a modular form [f, g]n of weight k +l + 2 n. In the present paper we study these "Rankin-Cohen brackets" from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure ("RC algebra") consisting of a graded vector space together with a collection of bilinear operations [,]n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivation S of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - φ{symbol}E, Φ - φ{symbol}2 + ∂(φ{symbol})) where φ{symbol} is an element of degree 2 and E is the Fuler operator (= multiplication by the degree). © 1994 Indian Academy of Science.