12 research outputs found
On uniformization of Burnside's curve
Main objects of uniformization of the curve 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 -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.Comment: Final version. LaTeX, 23 pages, 1 figure. The handbook for elliptic
functions has been moved to arXiv:0808.348