Surfaces de Riemann compactes et formule de trace d'Eichler
In this thesis, we will study several algebraic, geometrical and topological properties of compact Riemann surfaces.  Two principal subjects will be treated.  First, using the fact that every compact Riemann surfaces of genus g greater or equal to 2 has a finite number of Weierstrass points, we will be able to prove that those surfaces have a finite number of automorphism.  Afterward, we will study the Eichler's trace formula. This formula allow us to find the character of an automorphism acting on the space of holomorphic q-differentials.  We will start our study using Klein's quartic curve. We will apply Eichler's formula in this case, which will allow us to familiarize ourselves with the statement of the theorem.  Finally, we will demonstrate the Eichler's trace formula, treating the case where the automorphism acts fixed point freely separately from the case where the automorphism has fixed points.
Character Klein's curve Riemann surfaces Eichler's trace formula Automorphism