# Automorphic Schwarzian equations

@article{Sebbar2020AutomorphicSE,
title={Automorphic Schwarzian equations},
author={Abdellah Sebbar and Hicham Saber},
journal={Forum Mathematicum},
year={2020},
volume={32},
pages={1621 - 1636}
}
• Published 2 February 2020
• Mathematics
• Forum Mathematicum
Abstract This paper concerns the study of the Schwartz differential equation {h,τ}=s⁢E4⁡(τ){\{h,\tau\}=s\operatorname{E}_{4}(\tau)}, where E4{\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL2⁡(ℤ){\operatorname{SL}_{2}({\mathbb{Z}})}. We do so using the theory of equivariant functions on the complex upper-half plane as well as an…
5 Citations
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## References

SHOWING 1-10 OF 36 REFERENCES
Schwarzian differential equations and Hecke eigenforms on Shimura curves
Abstract Let X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to
Equivariant Forms: Structure and Geometry
• Mathematics
• 2013
Abstract. In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of
The Schwarzian derivative and schlicht functions
It is customary to formulate the inequalities of the "Verzerrungssatz" type for analytic functions w—f(z), schlicht in the unit circle, with reference to a specific normalization. The two
Meromorphic solutions of a system of functional equations involving the modular group
The purpose of this paper is to study the meromorphic solutions F of the system of functional equations (1) F[T(z) ] = T[F(z)], for all T in the modular group M and z in the upperhalf plane. A
Equivariant functions and integrals of elliptic functions
• Mathematics
• 2012
In this paper, we introduce the theory of equivariant functions by studying their analytic, geometric and algebraic properties. We also determine the necessary and sufficient conditions under which
On Automorphic Functions
The paper begins with a brief account of Christoffel’s work on abelian integrals and its historical context and then describes the close connection between Riemann surfaces and Fuchsian groups. This
Elliptic Zeta Functions and Equivariant Functions
• Mathematics