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

The main objects of uniformization of the curve y...=x...-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. (ProQuest: ... denotes formulae/symbols omitted.)