Modular Linear Differential Operators and Generalized Rankin-Cohen Brackets

TL;DR

This paper provides a comprehensive framework for expressing modular linear differential operators of any order using Rankin-Cohen brackets, quasimodular forms, and higher derivatives, unifying various approaches in the theory.

Contribution

It introduces new uniform descriptions of MLDOs through higher Serre derivatives and extended Rankin-Cohen brackets, connecting them with quasimodular and almost holomorphic forms.

Findings

Expressed all MLDOs in terms of Rankin-Cohen brackets and their modifications.

Extended the theory of quasimodular forms on various subgroups of SL_2(R).

Sharpened existing theorems on Rankin-Cohen brackets of quasimodular forms.

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