Modular Linear Differential Operators and Generalized Rankin-Cohen Brackets

Abstract

The aim in this paper is to give expressions for modular linear differential operators of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on the theory of quasimodular forms on both cocompact and non-cocompact subgroups of $SL_2(\mathbb{R})$, as well as a slight sharpening of a theorem of Martin and Royer on Rankin-Cohen brackets of quasimodular forms