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Abstract

Let~$X=\Po/\Gamma$ be an~$n$-punctured sphere, $n>3$. We introduce and
study~$n-3$ deformation operators on the space of modular forms~$M_*(\Gamma)$
based on the classical theory of uniformizing differential equations and
accessory parameters. When restricting to modular functions, we recover a
construction in Teichm\"uller theory related to the deformation of the complex
structure of~$X$. We describe the deformation operators in terms of derivations
with respect to Eichler integrals of weight-four cusp forms, and in terms of
vector-valued modular forms attached to extensions of symmetric tensor
representations.