On uniformization of Burnside's curve y2=x5−xy^2=x^5-x

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.Comment: Final version. LaTeX, 23 pages, 1 figure. The handbook for elliptic functions has been moved to arXiv:0808.348

Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \mathbb {R}^{3}, as hyperelliptic curves, and as \mathbb {CP}^{1} modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller

Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and \break Kadomtsev-Petviashvili (KP) equations were constructed for a given curve $y^2 = f(x)$ whose genus is three. This study was based upon the fact that about one hundred years ago (Acta Math. (1903) {\bf{27}}, 135-156), H. F. Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator ${\bold{D}}$, identities of Pfaffians, symmetric functions, hyperelliptic $\sigma$-function and $\wp$-functions; $\wp_{\mu \nu} = -\partial_\mu \partial_\nu \log \sigma$ $= - ({\bold{D}}_\mu {\bold{D}}_\nu \sigma \sigma)/2\sigma^2$. The connection between his theory and the modern soliton theory was also discussed.Comment: AMS-Tex, 12 page

On computing Belyi maps

We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French abstract; revised according to referee's suggestion

Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP solutions

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristic quantities of the Riemann surface with high precision even in almost degenerate situations as in the solitonic limit where the branch points coincide pairwise. As an example we consider hyperelliptic solutions to the Kadomtsev-Petviashvili and the Korteweg-de Vries equation. Tests of the numerics using identities for periods on the Riemann surface and the differential equations are performed. It is shown that an accuracy of the order of machine precision can be achieved.Comment: 16 pages, 8 figure