Higher Schwarzian, quasimodular forms and equivariant functions

TL;DR

This paper explores higher-order Schwarzian derivatives, revealing their relationship with quasimodular forms and equivariant functions, and extends classical results in complex analysis and modular forms.

Contribution

It establishes a characterization of equivariant functions via higher Schwarzians as quasimodular forms, linking differential operators to modular structures.

Findings

Higher Schwarzians are equivalent to quasimodular forms of specific weight and depth.

Characterization of equivariant functions through higher Schwarzians.

Extension of classical results connecting differential operators and modular forms.

Abstract

The Schwarzian derivative plays a fundamental role in complex analysis, differential equations, and modular forms. In this paper, we investigate its higher-order generalizations, known as higher Schwarzians, and their connections to quasimodular forms and equivariant functions. We prove that a meromorphic function is equivariant if and only if its higher Schwarzians are quasimodular forms of prescribed weight and depth, thereby extending classical results and linking projective differential operators to the structure of modular and quasimodular forms.