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

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

2001-11-13

arxiv.org/abs/math/0111150v5

Journal Math. Phys. (2009), v.50(10)

Mathematics

Classical Analysis and ODEs

Final version. LaTeX, 23 pages, 1 figure. The handbook for elliptic functions has been moved to arXiv:0808.3486

Scientific paper

10.1063/1.3215981

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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