I've taken a small course in Riemann surfaces, and there is one part that I still don't understand (and I've been unable to find a reference that explains this rigorously and in detail).
It is about the construction of a Riemann surface associated to an analytic germ.
By analytic germ we mean a couple $(z_0, (a_n)_{n \in \mathbb{N}})$ where $z_0$ and the $a_i$ are in the complex plane, and the series $\sum a_n (z-z_0)^n$ has strictly positive (but typically finite) radius of convergence.
Then (without getting into technical details) the Riemann surface associated to such a germ is the connected component of that germ in the space of all analytic germs, equipped with a certain topology. I get that it defines a Riemann surface.
I also understand "intuitively" what it does on the classic examples : for example, the Riemann surface associated to $(1,\log)$ is a sort of "spiral surface", since when we turn around zero, we add (if turning counter-clockwise) a $2 i \pi$ to the principal determination (so there are countably many sheaves).
In the case of the germ of the square root at $z_0=1$, it is a two-sheaves surface.
However I have no idea how one would go to determine this properly. In this example, I know that this surface is biholomorphic to $\mathbb{C}$ in the case of the log but I have no idea how to prove that.
Can anyone either give me a detailed reference or a sketch of the proof (but with all the key arguments) ?