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

Abstract

Let $X = \mathbb{H}/\Gamma$ be an $n$-punctured sphere, $n \gt 3$. We introduce and study $n-3$ deformation operators on the space of modular forms $M_\ast (\Gamma)$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichmüller 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.