On Uniformization of Algebraic Curves

Abstract

Based on Burnside's parametrization of the algebraic curve y 2 = x 5 − x we obtain remaining attributes of its uniformization: associated Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons, etc. As a generalization, we propose a way of uniformization of arbitrary curves by zero genus groups. In the hyperelliptic case all the objects of the theory are explicitly described. We consider a large number of examples and, briefly, applications: Abelian integrals, metrics of Poincaré, differential equations of the Jacobi-Chazy and Picard-Fuchs type, and others.

A τ-function solution of the sixth painlevé transcendent

“…if we use the substitution x = u 4 and y = −u 2 v. This curve does not belong to the series of classic modular equations [7] nor to their closest variations [12]. In turn, it is remarkable that because of the birational transformation…”

“…In those papers, solutions were also presented in the parametric forms and contained the set of functions θ, θ , θ , θ , ϑ, ϑ , and ϑ , although we can obtain a solution in a compact form if we transform Weierstrass functions (9) into the theta functions (cf. [12]). (7) is…”

“…Explicit expansions in the vicinities of all other points can be written using the formulas in the appendix. Formulas (12), (15), and (17) themselves provide an analytic answer for the power-law expansions in [4]. …”

“…For instance, in the above examples, besides the integrals K, K , E, E , i.e., functions of the type 2 F 1 (±1/2, 1/2; 1|z), we encounter the function η 2 (τ ). It can be shown [12] that the hypergeometric equation related to this function is just the equation for the function 2 F 1 (1/12, 1/12; 2/3|J), i.e., the equations that determine the classic Klein modular invariant J = (g 3 2 /∆)(τ ) [6], [7]:…”

“…(see [12] for the details; applications, albeit in a concise form, can also be found there). If we consider analogous solution (9) with the same parameters Aτ + B = 1/3, then we obtain the curve that is the Hitchin solution,…”