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

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.

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

References in corpus (1)

• On the uniformization of algebraic curves

Cited by in corpus (2)

• The sixth Painleve transcendent and uniformization of algebraic curves
• Construction of Hyperbolic Signal Sets from the Uniformization of Hyperelliptic Curves