Résumé
The uniformization theorem of Poincaré-Koebe states that any smooth compact Riemann surface of genus $g>1$<br /> is a quotient of the upper half-plane by a Fuchsian<br />group. On the other hand, a Riemann surface is also a complex<br />algebraic curve. In genus 2 and 3, these curves can always be<br />realized as plane curves, i.e as the set of zeros of a homogeneous<br />polynomial equation $P(x,y,z)=0$ with complex coefficients.<br /><br />In this thesis we deal with the explicit link between these two<br />descriptions for surfaces of genus 2 and 3 with non-trivial automorphisms.<br /><br />In genus 2, we first deal with surfaces having a non-trivial<br />involution. We describe the correspondence between the actions of two<br />groups, the first acting on the algebraic structures, and the second on<br />the hyperbolic structures of these surfaces. The relation between<br />these two groups enables us to interpret in terms of Dehn twists and<br />half-twists the links between the covers branched over the<br />same five distinct points of $\mathbb{P}^1(\mathbb{C})$. In particular, the<br />action of some Dehn twists can be read on the equations.<br />A similar study is done for surfaces having an order 3 automorphism.<br />We then study special algebraic families, in which the surfaces are<br />defined by a smaller number of parameters than those of the ambient spaces (but<br />not having necessarily more automorphisms).<br />We then deal with real families. We show in particular that the<br />various groups enable us to describe the algebraic and geometric links<br />between surfaces whose real components have different topological types.<br /><br />In genus 3, we study the relations between the equations of<br />the four genus 3 double<br />covers of a genus 1 curve branched over four given points. We also<br />describe the relations between their hyperbolic structure when they<br />are tiled by two right-angled hyperbolic hexagons.