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Let~$X=\Po/\Gamma$ be an~$n$-punctured sphere, $n>3$. We introduce andstudy~$n-3$ deformation operators on the space of modular forms~$M_*(\Gamma)$based on the classical theory of uniformizing differential equations andaccessory parameters. When restricting to modular functions, we recover aconstruction in Teichm\"uller theory related to the deformation of the complexstructure of~$X$. We describe the deformation operators in terms of derivationswith respect to Eichler integrals of weight-four cusp forms, and in terms ofvector-valued modular forms attached to extensions of symmetric tensorrepresentations.

Modular forms, deformation of punctured spheres, and extensions of symmetric tensor representations