Modular differential equations of second order with regular singularities at elliptic points for $SL_2(\mathbb {Z})$

Abstract

We give a definition of the modular differential equations of weight k for a discrete subgroup for Γ ⊂ SL 2 (R); in this paper we set Γ = SL 2 (Z). We solve such equations admitting regular singularities at elliptic points for SL 2 (Z) in terms of the Eisenstein series and the Gauss hypergeometric series. Furthermore, we give a series of such modular differential equations parametrized by an even integer k, and discuss some properties of solution spaces. We find several equations which are solved by a modular form of weight k.