Multilinear Operators on Siegel Modular Forms of Genus 1 and 2

Abstract

Classically, there are many interesting connections between differential operators and the theory of elliptic modular forms and many interesting results have been explored. In particular, it has been known for some time how to obtain an elliptic modular form from the derivatives ofN elliptic modular forms, which has already been studied in detail by R. Rankin in [9] and [10]. When N = 2, as a special case of Rankin’s result in [10], H. Cohen constructed certain covariant bilinear operators which he used to obtain modular forms with interesting Fourier coefficients[4]. Later, these covariant bilinear operators were called Rankin-Cohen operators by D. Zagier who studied their algebraic relations[12, 13] as well as connections with formal pseudodifferential operators[5]. Furthermore, Rankin-Cohen operators are shown to appear as the various terms in the (convergent) expansion of the composition of two symbols in a certain symbolic calculus associated with SL(2,R)[11]. Recently, the Rankin-Cohen type bracket operator on Jacobi forms and Siegel modular forms has been studied using the heat operator and differential operator, respectively. In fact, it was shown how to construct, explicitly, Jacobi forms and Siegel modular forms of genus 2 with the Rankin-Cohen type bracket operators involving the heat operator and the determinant-differential operators, respectively[1].