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

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.

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Received 23 August 2021

Received revised 7 October 2021

Accepted 19 October 2021

Published 14 May 2024