One of the consequences of the uniformization theorem of Koebe and Poincaré is that any smooth complex algebraic curve $C$ of genus $g > 1$ is conformally equivalent to ${\bb H}/G$, where $G \subset \hbox{PSL}_2({\bb R})$ is a Fuchsian group and is naturally endowed with a hyperbolic metric. Conversely, any compact hyperbolic 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.