Geodesics, Periods, and Equations of Real Hyperelliptic Curves

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. 0. Introduction The uniformization theorem of Koebe and Poincaré states that any Riemann surface has a universal covering conformally equivalent to either the Riemann sphere P1, the complex plane C, or the Poincaré upper half-plane H. One of the consequences is that any smooth complex algebraic curve C of genus g > 1 is conformally equivalent to H/G, where G ⊂ PSL2(R) is a Fuchsian group. Conversely, any compact Riemann surface is isomorphic to an algebraic curve. Hence, any curve of genus g > 1 may be described in two ways, either by an equation or by a Fuchsian group. Going explicitly from one description to the other is, in either direction, a difficult problem. This is the classical uniformization problem. In this paper we study the direction from the Fuchsian groups to the curves. We also provide a large number of new examples, all in genus 2, where the correspondence is given in exact form. Since H has a natural hyperbolic metric and this induces one on the corresponding curve C, one can reformulate the problem by asking how one relates explicitly DUKE MATHEMATICAL JOURNAL Vol. 108, No. 2, c © 2001 Received 1 August 2000. Revision received 18 October 2000. 2000 Mathematics Subject Classification. Primary 30F10; Secondary 14H15, 32G15. Authors’ work supported in part by European Community Human Capital and Mobility Programme plan number CHRX-CT93-0408 and by Swiss National Science Foundation contract numbers 21-50847.97, 21-57251.99.