On modular linear differential operators and their applications

TL;DR

This paper develops the algebraic framework of modular linear differential operators, explores their solution spaces, and establishes bounds on MLDEs satisfied by modular forms like Eisenstein series.

Contribution

It introduces a formal algebraic structure for modular linear differential operators and analyzes their properties and applications to solutions of MLDEs.

Findings

Solutions to MLDEs form an algebraic structure.

Any quasimodular form of weight k and depth s solves a monic MLDE of weight k-s.

A lower bound for the order of MLDEs satisfied by ${E_4}^m{E_6}^n$ is established.

Abstract

A formal definition of the graded algebra $\mathcal{R}$ of modular linear differential operators is given and its properties are studied. An algebraic structure of the solutions to modular linear differential equations (MLDEs) is shown. It is also proved that any quasimodular form of weight $k$ and depth $s$ becomes a solution to a monic MLDE of weight $k-s$. By using the algebraic properties of $\mathcal{R}$, linear differential operators which map the solution space of a monic MLDE to that of another are determined for sufficiently low weights and orders. Furthermore, a lower bound of the order of monic MLDEs satisfied by ${E_4}^m{E_6}^n$ is found.