Here is a question which has been bothering me for a week. It comes from Otto Forster's lectures on Riemann surfaces. The question comes from Exercise 10.1(c):
Let $X$ be a riemann surface and $\omega$ a holomorphic 1-form on $X$. If $\varphi$ is a local primitive of $\omega$ in a neighbourhood of a point $a\in X$. Let $(Y,p,f,b)$ be a maximal analytic continuation of $\varphi$. Show that
(c) The covering $p:Y\rightarrow X$ is Galois and Deck(Y/X) is abelian.
I can't show that Deck(Y/X) is abelian. In fact, what I have tried is as follows: I let $t:\tilde{X}\rightarrow X$ be the universal covering of $X$, and $\tilde{F}$ the primitive of $t^{*}\omega$. Then we have a group homomorphism
$\tau:\mbox{Aut}(\tilde{X}\rightarrow X)\rightarrow(\mathbb{C},+)$,
given by
$\sigma\mapsto F-\sigma F=F-F\circ\sigma^{-1}:=a_{\sigma}$.
I am done if the kernel of this map is $\mbox{Aut}(\tilde{X}\rightarrow Y)$, for then
$\mbox{Aut}(Y\rightarrow X)=\frac{\mbox{Aut}(\tilde{X}\rightarrow X)}{\mbox{Aut}(\tilde{X}\rightarrow Y)}=\frac{\mbox{Aut}(\tilde{X}\rightarrow X)}{\mbox{ker}\tau}$
and hence is abelian.
I have managed to show that $\mbox{Aut}(\tilde{X}\rightarrow Y)\subseteq\ker\lambda$, but the obstruction lies in the reverse inclusion.
Perhaps someone can me some hint on how to proceed, or to give a completely new idea to this question.
Thanks!
Opps, in fact the $Aut(A\rightarrow B)$ is actually $\mbox{Deck}(A/B)$ because I am using the French notation for deck transformation.
Since any two local primitives of $\omega$ differ by a constant, analytic continuation along closed curves defines a homomorphism $\psi\colon\pi_1(X) \to (\mathbb{C},+)$. Let $K$ be the kernel of this homomorphism. Then $Y$ is the cover of $X$ corresponding to $K$, and $\mathrm{Deck}(Y/X)$ is isomorphic to the quotient $\pi_1(X)/K$. This quotient is isomorphic to the image of $\psi$, which is abelian since it's a subgroup of $(\mathbb{C},+)$.