Geodesics, periods, and equations of real hyperelliptic curves

Abstract

In this paper we start a new approach to the uniformization problem of Riemann surfaces and algebraic curves by means of computational procedures. The following question is studied: Given a compact Riemann surface S described as the quotient of the Poincaré upper half-plane divided by the action of a Fuchsian group, find explicitly the polynomial describing S as an algebraic curve (in some normal form). The explicit computation given in this paper is based on the numerical computation of conformal capacities of hyperbolic domains. These capacities yield the period matrices of S in terms of the Fenchel-Nielsen coordinates, and from there one gets to the polynomial via theta-characteristics. The paper also contains a list of worked-out examples and a list of examples–new in the literature–where the polynomial for the curve, as a function of the corresponding Fuchsian group, is given in closed form.