FILIPUK, GALINA

VIDUNAS, RAIMUNDAS

Osaka City University Repository

A CLASSIFICATION OF COVERINGS YIELDING HEUN-TO-HYPERGEOMETRIC REDUCTIONS

Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Möbius automorphisms, the coverings P1→ P1 that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases with Liouvillian solutions). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 are compositions of smaller degree transformations between hypergeometric and Heun functions. The 61 transformations are realized by 48 different Belyi coverings (though 2 coverings should be counted twice as their moduli field is quadratic). 38 of these coverings appear in Herfurtner’s list of elliptic surfaces over P1 with four singular fibers, as their j -invariants. In passing, we show in an elegant way that there are no coverings with some branching patterns

Vidunas, Raimundas

Filipuk, Galina

Osaka University Knowledge Archive

http://ir.library.osaka-u.ac.jp/dspace/bitstream/11094/50975/1/ojm51_04_867.pdf

A CLASSIFICATION OF COVERINGS YIELDING HEUN-TO-HYPERGEOMETRIC REDUCTIONS

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