## Duke Mathematical Journal

- Duke Math. J.
- Volume 108, Number 2 (2001), 211-250.

### 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.

#### Article information

**Source**

Duke Math. J., Volume 108, Number 2 (2001), 211-250.

**Dates**

First available in Project Euclid: 5 August 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1091737156

**Digital Object Identifier**

doi:10.1215/S0012-7094-01-10822-3

**Mathematical Reviews number (MathSciNet)**

MR1833391

**Zentralblatt MATH identifier**

1019.30042

**Subjects**

Primary: 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15]

Secondary: 14P05: Real algebraic sets [See also 12D15, 13J30] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]

#### Citation

Buser, Peter; Silhol, Robert. Geodesics, periods, and equations of real hyperelliptic curves. Duke Math. J. 108 (2001), no. 2, 211--250. doi:10.1215/S0012-7094-01-10822-3. https://projecteuclid.org/euclid.dmj/1091737156