Automorphisms of Riemann surfaces

Abstract

[en] In the middle of the 19th century, Bernhard Riemann began to develop the idea of the today known as Riemann Surfaces, with the aim of constructing the theory of analytic functions of complex variables in a more solid way. Similarly as in differential geometry we develop a theory of manifolds that locally looks like $\mathbb{R}^n$, we can develop an analogous theory about manifolds that locally looks like $\mathbb{C}^n$. In other words, Riemann Surfaces are manifolds of complex dimension one and we can consider them as curves. Treated as a curve, we could be interested in studying the automorphisms of compact and connected Riemann Surfaces. Already at the beginning of the 19th century, it was shown that the curves of genus $g=0$ and $g=1$ had an infinite number of automorphisms. However, it remained to be seen what happened for curves with $g \geq 2$. It was not until 1878 that Schwarz proved that for $g \geq 2$ the group of automorphisms was finite (Schwarz's Theorem). Later, in 1893, Hurwitz went further and gave an upper bound of $84(g-1)$ for the number of automorphisms, namely Hurwitz's Theorem. Thus, our principal objective in this work has been to prove Hurwitz's Theorem. In order to do it, we have required several objects that have themselves much importance and we could dedicate an entire paper for each of them.

Before describing how will be the work organized let's give an idea of how we will proceed. We will see that the group of automorphisms of $X$ (compact and connected Riemann Surface) permutes Weierstrass points $(\mathrm{W}(\mathrm{X})$ ) which we will prove are finite using the Wronskian. Once we will have described $A u t(X)$ and $\operatorname{Perm}(W(X))$ we will define the morphism $\lambda: \operatorname{Aut}(X) \rightarrow \operatorname{Perm}(W(X))$ which is (0) or $\mathbb{Z} / 2$ and prove that $\operatorname{Aut}(X)$ is a finite group. After having proved the finiteness of $\operatorname{Aut}(X)$, we will go to the quotient $X / \operatorname{Aut}(X)$ and using the RiemannHurwitz formula find the bound of the Hurwitz Theorem.