Modular differential equations of second order with regular singularities at elliptic points for 𝑆𝐿₂(ℤ)

Tsutsumi, Hiroyuki

doi:10.1090/S0002-9939-05-08115-3

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

Proc. Amer. Math. Soc. 134 (2006), 931-941 Request permission

Abstract:

We give a definition of the modular differential equations of weight $k$ for a discrete subgroup for $\Gamma \subset SL_2(\mathbb {R})$; in this paper we set $\Gamma = SL_2(\mathbb {Z})$. We solve such equations admitting regular singularities at elliptic points for $SL_2(\mathbb {Z})$ 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 $k$.

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