# Tagged Questions

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### The number of holomorphic coverings (with given degree) of the punctured sphere is finite.

I'm looking for a proof of the following theorem: Fix a finite set $B=\{y_1,\ldots,y_k\}\subseteq \mathbb P^1(\mathbb C)$, then there is only a finite number of isomoprhism classes of ...
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### Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
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### Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
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### The covering space of a region contained in complex plane delete two points.

We all know that C \ {0,1} can be given the Poincare hyperbolic metric, so that a region W in it is an embedded manifold of negative constant curvature. Hence the covering space of W is a hyperbolic ...
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### Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
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### An entire function with finite covering group is a polynomial.

Let $f$ be an entire function. Think of it as a covering space of $\mathbb{C}$ (perhaps with isolated punctures) to $\mathbb{C}$ (perhaps with isolated punctures). Suppose we know there is only a ...
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### Constructing Riemann surfaces using the covering spaces

In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow: A polynomail-like map ...
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### Why is this covering map doubly periodic?

The universal cover of the torus $T$ is the complex plane $\mathbb{C}$. If $p: \mathbb{C} \to T$ is the covering map, why is $p$ doubly periodic?
I'm trying to work through the exercises in Otto Forster's book on Riemann Surfaces. While most of them seemed not that hard, this one gives me a headache: Let $X=\mathbb{C}\setminus\{\pm1\}$ and $Y ... 1answer 196 views ### Branch points of rational functions Let$f$be a rational function on a compact connected Riemann surface$X$. The rational function$f$induces a holomorphic map$\overline{f}:X\to \mathbf{P}^1(\mathbf{C})$. Let$x$be a point on the ... 0answers 106 views ### Very Basic Covering Space Q. - Two different covering maps over a same space? Here's the situation I'm in. I have a map from the (closure of) Upper Half Plane ($\mathbb{U}$) into the punctured (closed) disk ($\mathbb{D}$) called$q$that satisfies$q(0) = 1$, and$q(z+1) = ...
Let $f:X\to \mathbf{P}^1$ be a simple cover of the Riemann sphere. This means that $f$ is a branched cover, and that each fibre has at least $\deg f-1$ points in it. Is it true that the number of ...