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On uniformization of Burnside's curve $y^2=x^5-x$

Y. V. Brezhnev
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Abstract

Main objects of uniformization of the curve $y^2=x^5-x$ are studied: its Burnside's parametrization, corresponding Schwarz's equation, and accessory parameters. As a result we obtain the first examples of solvable Fuchsian equations on torus and exhibit number-theoretic integer $q$-series for uniformizing functions, relevant modular forms, and analytic series for holomorphic Abelian integrals. A conjecture of Whittaker for hyperelliptic curves and its hypergeometric reducibility are discussed. We also consider the conversion between Burnside's and Whittaker's uniformizations.

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BibTeXRIS

Y. V. Brezhnev. 2009-12-17. On uniformization of Burnside's curve $y^2=x^5-x$. https://doi.org/10.1063/1.3215981

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