Uniformizing differential equations of arithmetic (1;e)-groups

TL;DR

This paper introduces a numerical method to compute accessory parameters of uniformizing differential equations for elliptic curves linked to arithmetic Fuchsian (1;e)-groups, providing a complete classification up to conjugation.

Contribution

The paper develops a numerical approach to determine accessory parameters and offers a complete list of these parameters for all relevant arithmetic (1;e)-groups.

Findings

Numerical method successfully computes accessory parameters.

Complete classification of (1;e)-groups up to conjugation.

Provides explicit accessory parameters for all cases.

Abstract

We present a numerical method to compute accessory parameters of the uniformizing differential equations of elliptic curves associated to arithmetic Fuchsian $(1;e)$ groups. Up to $PSL_2(\bbr)$ conjugation there are only finitely many such groups and we give a complete list of the accessory parameters