On modular linear differential operators and their applications

2018/07/19 by Yamashita, Fumitoshi
#16W50 #34G10 #FOS: Mathematics #Number Theory (math.NT) #Primary 11F11 #Secondary 13N10

paper · doi:10.48550/arxiv.1807.07204

Abstract

A formal definition of the graded algebra 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 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 E₄^mE₆ⁿ is found.

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