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Belyi functions for hyperbolic hypergeometric-to-Heun transformations. (English) Zbl 1332.33036

Belyi functions and “dessin d’enfants” is a motivating field of research in algebraic geometry, complex analysis, Galois theory and related fields. However, computations of Belyi functions of degree over 20 is still considered hard even for genus on Belyi coverings \(\mathbb{P}^1\to\mathbb{P}^1\). The subject of this paper is effective computation of certain Belyi functions \(\mathbb{P}^1\to\mathbb{P}^1\), of degree up to 60, utilizing the fact that those functions transform Fuchsian differential equations with a small number of singularities. The exact results are too involved to be stated here.

MSC:

33E30 Other functions coming from differential, difference and integral equations
11G32 Arithmetic aspects of dessins d’enfants, Belyĭ theory
14H57 Dessins d’enfants theory
34M03 Linear ordinary differential equations and systems in the complex domain
33C05 Classical hypergeometric functions, \({}_2F_1\)
57M12 Low-dimensional topology of special (e.g., branched) coverings
14-04 Software, source code, etc. for problems pertaining to algebraic geometry

Software:

ComputeBelyi
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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