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Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.

Q1. (Algebraic formulation) Does there exist a finite (surjective) morphism `$\pi:C\longrightarrow \mathbf{P}^1_{\bar{\mathbf{Q}}}$` such that $\pi|_{U}$ is etale?

Equivalently, let $X$ be a compact connected Riemann surface of genus $g$ which can be defined over $\bar{\mathbf{Q}}$ and let $B$ be a finite set of of closed points in $X$ with complement $Y$.

Q1. (Analytic formulation ) Does there exist a finite topological cover `$Y\longrightarrow \mathbf{P}^1(\mathbf{C})-\{0,1,\infty\}$` ?

The equivalence of these two questions follows from the proof of Belyi's theorem and Riemann's existence Theorem.

If the answer to Question 1 is positive, I would be very interested in knowing if the degree of $\pi$ can be bounded effectively.

Q2. Does there exist a finite (surjective) morphism $\pi:C\longrightarrow \mathbf{P}^1$ such that $\pi|_{U}$ is etale and $\deg \pi \leq c$, where $c$ is a constant depending only on $S$ and $g$?

Example. Suppose that $g=0$. Then, following Belyi's proof of his theorem, the answer to Question 1 is yes. The answer to Question 2 is also positive and an explicit upper bound for such a rational function is given by Khadjavi in An effective version of Belyi's Theorem.

I don't expect the answer to Question 1 to be easy. In fact, what I'm asking is to prove the existence of a Belyi morphism `$\pi:C\longrightarrow \mathbf{P}^1_{\bar{\mathbf{Q}}}$` with prescribed ramification. Now, that's probably very hard but definitely very interesting to find out.

Trivial Remark. Suppose that $g>1$. Then the automorphism group of $C$ is finite. Choose a Belyi morphism `$\pi:C\longrightarrow \mathbf{P}^1_{\bar{\mathbf{Q}}}$` and let $U_0\subset C$ be the complement of the ramification points of $\pi$. Then we see that Question 1 has a positive answer if we take $U$ to be $\sigma(U_0)$ with $\sigma$ an automorphism of $C$. But that's only finitely many examples.

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# Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar

Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.

Q1. (Algebraic formulation) Does there exist a finite (surjective) morphism `$\pi:C\longrightarrow \mathbf{P}^1_{\bar{\mathbf{Q}}}$` such that $\pi|_{U}$ is etale?

Equivalently, let $X$ be a compact connected Riemann surface of genus $g$ which can be defined over $\bar{\mathbf{Q}}$ and let $B$ be a finite set of of closed points in $X$ with complement $Y$.

Q1. (Analytic formulation ) Does there exist a finite topological cover `$Y\longrightarrow \mathbf{P}^1(\mathbf{C})-\{0,1,\infty\}$` ?

The equivalence of these two questions follows from the proof of Belyi's theorem and Riemann's existence Theorem.

If the answer to Question 1 is positive, I would be very interested in knowing if the degree of $\pi$ can be bounded effectively.

Q2. Does there exist a finite (surjective) morphism $\pi:C\longrightarrow \mathbf{P}^1$ such that $\pi|_{U}$ is etale and $\deg \pi \leq c$, where $c$ is a constant depending only on $S$ and $g$?

Example. Suppose that $g=0$. Then, following Belyi's proof of his theorem, the answer to Question 1 is yes. The answer to Question 2 is also positive and an explicit upper bound for such a rational function is given by Khadjavi in An effective version of Belyi's Theorem.

I don't expect the answer to Question 1 to be easy. In fact, what I'm asking is to prove the existence of a Belyi morphism `$\pi:C\longrightarrow \mathbf{P}^1_{\bar{\mathbf{Q}}}$` with prescribed ramification. Now, that's probably very hard but definitely very interesting to find out.

Trivial Remark. Suppose that $g>1$. Then the automorphism group of $C$ is finite. Choose a Belyi morphism `$\pi:C\longrightarrow \mathbf{P}^1_{\bar{\mathbf{Q}}}$` and let $U_0\subset C$ be the complement of the ramification points of $\pi$. Then we see that Question 1 has a positive answer if we take $U$ to be $\sigma(U_0)$ with $\sigma$ an automorphism of $C$. But that's only finitely many examples.
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