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arXiv · 2002.00493

Automorphic Schwarzian equations

Abdellah Sebbar·Hicham Saber
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Abstract

This paper concerns the study of the Schwarz differential equation $\{h,\tau \}=s\,E_4(\tau)$ where $E_4$ 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 $\mbox{SL}_2(\mathbb Z)$. We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of $\mbox{SL}_2(\mathbb Z)$. This also leads to the solutions to the Fuchsian differential equation $y''+s\,E_4\,y=0$.

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Abdellah Sebbar, Hicham Saber. 2020-02-02. Automorphic Schwarzian equations. https://arxiv.org/abs/2002.00493

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