Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 024, 20 pages      arXiv:2004.02971      https://doi.org/10.3842/SIGMA.2022.024

Accessory Parameters for Four-Punctured Spheres

Gabriele Bogo
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany

Received August 24, 2021, in final form March 22, 2022; Published online March 28, 2022

Abstract
We study the accessory parameter problem for four-punctured spheres from the point of view of modular forms. The value of the accessory parameter giving the uniformization is characterized as the unique zero of a system of equations. This gives an effective method to compute the uniformizing differential equation. As an application, we compute numerically and study the local expansion of the real-analytic function associating to a four-punctured sphere the value of its uniformizing parameter, and make some observations on its coefficients.

Key words: accessory parameters; Fuchsian uniformization; modular forms.

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