Higher Schwarzian, quasimodular forms and equivariant functions
Hicham Saber, Abdellah Sebbar
TL;DR
The paper addresses how higher Schwarzian derivatives generalize the connection between projective differential operators and modular-analytic structures in the setting of equivariant functions. It develops higher Schwarzians $S_n[f]$ via Aharonov invariants, analyzes their behavior under Möbius transformations, and proves that these objects are quasimodular forms of weight $2n$ and depth $n-2$ precisely when the underlying meromorphic function $f$ is $\rho$-equivariant for some representation $\rho$ of a discrete group $\Gamma$. The main contributions are a complete equivalence between $\rho$-equivariance and the quasimodularity of all $S_n[f]$ for $n\ge2$, a demonstration that $S_2[f]$ recovers the classical Schwarzian as a weight-4 modular form, and a structural account of how these objects fit into the algebra of quasimodular forms with an $\mathfrak{sl}_2(\mathbb{C})$-action. This work unifies projective differential operators with modular and quasimodular form theory and has implications for monodromy representations, vector bundles on modular curves, and modular differential equations.
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.