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

Abstract

Let~$X=\Po/Γ$ be an~$n$-punctured sphere, $n>3$. 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.

15 pages. The result on differential equations has been improved. A new section on vector-valued modular forms and extension of symmetric tensor representations is included

References in corpus (1)

• Accessory Parameters for Four-Punctured Spheres